- For a new problem, you will need to begin a new live expert session.
- You can contact support with any questions regarding your current subscription.
- You will be able to enter math problems once our session is over.
- I am only able to help with one math problem per session. Which problem would you like to work on?
- Does that make sense?
- I am currently working on this problem.
- Are you still there?
- It appears we may have a connection issue. I will end the session - please reconnect if you still need assistance.
- Let me take a look...
- Can you please send an image of the problem you are seeing in your book or homework?
- If you click on "Tap to view steps..." you will see the steps are now numbered. Which step # do you have a question on?
- Please make sure you are in the correct subject. To change subjects, please exit out of this live expert session and select the appropriate subject from the menu located in the upper left corner of the Mathway screen.
- What are you trying to do with this input?
- While we cover a very wide range of problems, we are currently unable to assist with this specific problem. I spoke with my team and we will make note of this for future training. Is there a different problem you would like further assistance with?
- Mathway currently does not support this subject. We are more than happy to answer any math specific question you may have about this problem.
- Mathway currently does not support Ask an Expert Live in Chemistry. If this is what you were looking for, please contact support.
- Mathway currently only computes linear regressions.
- We are here to assist you with your math questions. You will need to get assistance from your school if you are having problems entering the answers into your online assignment.
- Phone support is available Monday-Friday, 9:00AM-10:00PM ET. You may speak with a member of our customer support team by calling 1-800-876-1799.
- Have a great day!
- Hope that helps!
- You're welcome!
- Per our terms of use, Mathway's live experts will not knowingly provide solutions to students while they are taking a test or quiz.

Please ensure that your password is at least 8 characters and contains each of the following:

- a special character: @$#!%*?&

- Announcements
- Information
- Unit 1 Relations & Functions
- Unit 2 Quadratics
- Unit 3: Polynomials
- Unit 4: Radical Functions
- Unit 5: Exponential & Log Functions
- Unit 6: Sequences & Series
- Unit 7: Probability
- Unit 8: Statistics
- Unit 9: Rational Functions
- Unit 10: Trigonometry
- Unit 2: Matrices
- Unit 3: Quadratics
- Unit 4: Polynomials
- Unit 5: Radical Functions
- Unit 6: Exponential & Log Functions
- Unit 7: Sequences & Series
- Unit 8: Probability
- Unit 9: Real World Probability
- Unit 10: Statistics
- Unit 11: Rational Functions

## Unit 12: Trigonometry

- Unit 4: Linear Systems
- Unit 5: Absolute Value & Piecewise Functions
- Unit 6: Exponential Functions
- Unit 7: Polynomials
- Unit 8: Quadratics
- Sharp Student of the Week
- About Ms. Sharp
- Math Problem of the Week
- Weekly Newsletter

If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

To log in and use all the features of Khan Academy, please enable JavaScript in your browser.

## Unit 1: Right triangles & trigonometry

About this unit, ratios in right triangles.

- Getting ready for right triangles and trigonometry (Opens a modal)
- Hypotenuse, opposite, and adjacent (Opens a modal)
- Side ratios in right triangles as a function of the angles (Opens a modal)
- Using similarity to estimate ratio between side lengths (Opens a modal)
- Using right triangle ratios to approximate angle measure (Opens a modal)
- Right triangles & trigonometry: FAQ (Opens a modal)
- Use ratios in right triangles Get 3 of 4 questions to level up!

## Introduction to the trigonometric ratios

- Triangle similarity & the trigonometric ratios (Opens a modal)
- Trigonometric ratios in right triangles (Opens a modal)
- Trigonometric ratios in right triangles Get 3 of 4 questions to level up!

## Solving for a side in a right triangle using the trigonometric ratios

- Solving for a side in right triangles with trigonometry (Opens a modal)
- Solve for a side in right triangles Get 3 of 4 questions to level up!

## Solving for an angle in a right triangle using the trigonometric ratios

- Intro to inverse trig functions (Opens a modal)
- Solve for an angle in right triangles Get 3 of 4 questions to level up!

## Sine and cosine of complementary angles

- Intro to the Pythagorean trig identity (Opens a modal)
- Sine & cosine of complementary angles (Opens a modal)
- Using complementary angles (Opens a modal)
- Trig word problem: complementary angles (Opens a modal)
- Trig challenge problem: trig values & side ratios (Opens a modal)
- Trig ratios of special triangles (Opens a modal)
- Relate ratios in right triangles Get 3 of 4 questions to level up!

## Modeling with right triangles

- Right triangle word problem (Opens a modal)
- Angles of elevation and depression (Opens a modal)
- Right triangle trigonometry review (Opens a modal)
- Right triangle trigonometry word problems Get 3 of 4 questions to level up!

## The reciprocal trigonometric ratios

- Reciprocal trig ratios (Opens a modal)
- Finding reciprocal trig ratios (Opens a modal)
- Using reciprocal trig ratios (Opens a modal)
- Trigonometric ratios review (Opens a modal)
- Reciprocal trig ratios Get 5 of 7 questions to level up!

- 7.4 The Other Trigonometric Functions
- Introduction to Prerequisites
- 1.1 Real Numbers: Algebra Essentials
- 1.2 Exponents and Scientific Notation
- 1.3 Radicals and Rational Exponents
- 1.4 Polynomials
- 1.5 Factoring Polynomials
- 1.6 Rational Expressions
- Key Equations
- Key Concepts
- Review Exercises
- Practice Test
- Introduction to Equations and Inequalities
- 2.1 The Rectangular Coordinate Systems and Graphs
- 2.2 Linear Equations in One Variable
- 2.3 Models and Applications
- 2.4 Complex Numbers
- 2.5 Quadratic Equations
- 2.6 Other Types of Equations
- 2.7 Linear Inequalities and Absolute Value Inequalities
- Introduction to Functions
- 3.1 Functions and Function Notation
- 3.2 Domain and Range
- 3.3 Rates of Change and Behavior of Graphs
- 3.4 Composition of Functions
- 3.5 Transformation of Functions
- 3.6 Absolute Value Functions
- 3.7 Inverse Functions
- Introduction to Linear Functions
- 4.1 Linear Functions
- 4.2 Modeling with Linear Functions
- 4.3 Fitting Linear Models to Data
- Introduction to Polynomial and Rational Functions
- 5.1 Quadratic Functions
- 5.2 Power Functions and Polynomial Functions
- 5.3 Graphs of Polynomial Functions
- 5.4 Dividing Polynomials
- 5.5 Zeros of Polynomial Functions
- 5.6 Rational Functions
- 5.7 Inverses and Radical Functions
- 5.8 Modeling Using Variation
- Introduction to Exponential and Logarithmic Functions
- 6.1 Exponential Functions
- 6.2 Graphs of Exponential Functions
- 6.3 Logarithmic Functions
- 6.4 Graphs of Logarithmic Functions
- 6.5 Logarithmic Properties
- 6.6 Exponential and Logarithmic Equations
- 6.7 Exponential and Logarithmic Models
- 6.8 Fitting Exponential Models to Data
- Introduction to The Unit Circle: Sine and Cosine Functions
- 7.2 Right Triangle Trigonometry
- 7.3 Unit Circle
- Introduction to Periodic Functions
- 8.1 Graphs of the Sine and Cosine Functions
- 8.2 Graphs of the Other Trigonometric Functions
- 8.3 Inverse Trigonometric Functions
- Introduction to Trigonometric Identities and Equations
- 9.1 Verifying Trigonometric Identities and Using Trigonometric Identities to Simplify Trigonometric Expressions
- 9.2 Sum and Difference Identities
- 9.3 Double-Angle, Half-Angle, and Reduction Formulas
- 9.4 Sum-to-Product and Product-to-Sum Formulas
- 9.5 Solving Trigonometric Equations
- Introduction to Further Applications of Trigonometry
- 10.1 Non-right Triangles: Law of Sines
- 10.2 Non-right Triangles: Law of Cosines
- 10.3 Polar Coordinates
- 10.4 Polar Coordinates: Graphs
- 10.5 Polar Form of Complex Numbers
- 10.6 Parametric Equations
- 10.7 Parametric Equations: Graphs
- 10.8 Vectors
- Introduction to Systems of Equations and Inequalities
- 11.1 Systems of Linear Equations: Two Variables
- 11.2 Systems of Linear Equations: Three Variables
- 11.3 Systems of Nonlinear Equations and Inequalities: Two Variables
- 11.4 Partial Fractions
- 11.5 Matrices and Matrix Operations
- 11.6 Solving Systems with Gaussian Elimination
- 11.7 Solving Systems with Inverses
- 11.8 Solving Systems with Cramer's Rule
- Introduction to Analytic Geometry
- 12.1 The Ellipse
- 12.2 The Hyperbola
- 12.3 The Parabola
- 12.4 Rotation of Axes
- 12.5 Conic Sections in Polar Coordinates
- Introduction to Sequences, Probability and Counting Theory
- 13.1 Sequences and Their Notations
- 13.2 Arithmetic Sequences
- 13.3 Geometric Sequences
- 13.4 Series and Their Notations
- 13.5 Counting Principles
- 13.6 Binomial Theorem
- 13.7 Probability
- A | Proofs, Identities, and Toolkit Functions

## Learning Objectives

In this section you will:

- Find exact values of the trigonometric functions secant, cosecant, tangent, and cotangent of π 3 , π 4 , π 3 , π 4 , and π 6 . π 6 .
- Use reference angles to evaluate the trigonometric functions secant, tangent, and cotangent.
- Use properties of even and odd trigonometric functions.
- Recognize and use fundamental identities.
- Evaluate trigonometric functions with a calculator.

A wheelchair ramp that meets the standards of the Americans with Disabilities Act must make an angle with the ground whose tangent is 1 12 1 12 or less, regardless of its length. A tangent represents a ratio, so this means that for every 1 inch of rise, the ramp must have 12 inches of run. Trigonometric functions allow us to specify the shapes and proportions of objects independent of exact dimensions. We have already defined the sine and cosine functions of an angle. Though sine and cosine are the trigonometric functions most often used, there are four others. Together they make up the set of six trigonometric functions. In this section, we will investigate the remaining functions.

## Finding Exact Values of the Trigonometric Functions Secant, Cosecant, Tangent, and Cotangent

We can also define the remaining functions in terms of the unit circle with a point ( x , y ) ( x , y ) corresponding to an angle of t , t , as shown in Figure 1 . As with the sine and cosine, we can use the ( x , y ) ( x , y ) coordinates to find the other functions.

The first function we will define is the tangent. The tangent of an angle is the ratio of the y -value to the x -value of the corresponding point on the unit circle. In Figure 1 , the tangent of angle t t is equal to y x , x ≠ 0. y x , x ≠ 0. Because the y -value is equal to the sine of t , t , and the x -value is equal to the cosine of t , t , the tangent of angle t t can also be defined as sin t cos t , cos t ≠ 0. sin t cos t , cos t ≠ 0. The tangent function is abbreviated as tan . tan . The remaining three functions can all be expressed as reciprocals of functions we have already defined.

- The secant function is the reciprocal of the cosine function. In Figure 1 , the secant of angle t t is equal to 1 cos t = 1 x , x ≠ 0. 1 cos t = 1 x , x ≠ 0. The secant function is abbreviated as sec . sec .
- The cotangent function is the reciprocal of the tangent function. In Figure 1 , the cotangent of angle t t is equal to cos t sin t = x y , y ≠ 0. cos t sin t = x y , y ≠ 0. The cotangent function is abbreviated as cot . cot .
- The cosecant function is the reciprocal of the sine function. In Figure 1 , the cosecant of angle t t is equal to 1 sin t = 1 y , y ≠ 0. 1 sin t = 1 y , y ≠ 0. The cosecant function is abbreviated as csc . csc .

## Tangent, Secant, Cosecant, and Cotangent Functions

If t t is a real number and ( x , y ) ( x , y ) is a point where the terminal side of an angle of t t radians intercepts the unit circle, then

## Finding Trigonometric Functions from a Point on the Unit Circle

The point ( − 3 2 , 1 2 ) ( − 3 2 , 1 2 ) is on the unit circle, as shown in Figure 2 . Find sin t , cos t , tan t , sec t , csc t , sin t , cos t , tan t , sec t , csc t , and cot t . cot t .

Because we know the ( x , y ) ( x , y ) coordinates of the point on the unit circle indicated by angle t , t , we can use those coordinates to find the six functions:

The point ( 2 2 , − 2 2 ) ( 2 2 , − 2 2 ) is on the unit circle, as shown in Figure 3 . Find sin t , cos t , tan t , sec t , csc t , sin t , cos t , tan t , sec t , csc t , and cot t . cot t .

## Finding the Trigonometric Functions of an Angle

Find sin t , cos t , tan t , sec t , csc t , sin t , cos t , tan t , sec t , csc t , and cot t . cot t . when t = π 6 . t = π 6 .

We have previously used the properties of equilateral triangles to demonstrate that sin π 6 = 1 2 sin π 6 = 1 2 and cos π 6 = 3 2 . cos π 6 = 3 2 . We can use these values and the definitions of tangent, secant, cosecant, and cotangent as functions of sine and cosine to find the remaining function values.

Find sin t , cos t , tan t , sec t , csc t , sin t , cos t , tan t , sec t , csc t , and cot t . cot t . when t = π 3 . t = π 3 .

Because we know the sine and cosine values for the common first-quadrant angles, we can find the other function values for those angles as well by setting x x equal to the cosine and y y equal to the sine and then using the definitions of tangent, secant, cosecant, and cotangent. The results are shown in Table 1 .

## Using Reference Angles to Evaluate Tangent, Secant, Cosecant, and Cotangent

We can evaluate trigonometric functions of angles outside the first quadrant using reference angles as we have already done with the sine and cosine functions. The procedure is the same: Find the reference angle formed by the terminal side of the given angle with the horizontal axis. The trigonometric function values for the original angle will be the same as those for the reference angle, except for the positive or negative sign, which is determined by x - and y -values in the original quadrant. Figure 4 shows which functions are positive in which quadrant.

To help remember which of the six trigonometric functions are positive in each quadrant, we can use the mnemonic phrase “A Smart Trig Class.” Each of the four words in the phrase corresponds to one of the four quadrants, starting with quadrant I and rotating counterclockwise. In quadrant I, which is “ A ,” underline a end underline ll of the six trigonometric functions are positive. In quadrant II, “ S mart,” only underline s end underline ine and its reciprocal function, cosecant, are positive. In quadrant III, “ T rig,” only underline t end underline angent and its reciprocal function, cotangent, are positive. Finally, in quadrant IV, “ C lass,” only underline c end underline osine and its reciprocal function, secant, are positive.

Given an angle not in the first quadrant, use reference angles to find all six trigonometric functions.

- Measure the angle formed by the terminal side of the given angle and the horizontal axis. This is the reference angle.
- Evaluate the function at the reference angle.
- Observe the quadrant where the terminal side of the original angle is located. Based on the quadrant, determine whether the output is positive or negative.

## Using Reference Angles to Find Trigonometric Functions

Use reference angles to find all six trigonometric functions of − 5 π 6 . − 5 π 6 .

The angle between this angle’s terminal side and the x -axis is π 6 , π 6 , so that is the reference angle. Since − 5 π 6 − 5 π 6 is in the third quadrant, where both x x and y y are negative, cosine, sine, secant, and cosecant will be negative, while tangent and cotangent will be positive.

cos ( − 5 π 6 ) = − 3 2 , sin ( − 5 π 6 ) = − 1 2 , tan ( − 5 π 6 ) = 3 3 , sec ( − 5 π 6 ) = − 2 3 3 , csc ( − 5 π 6 ) = −2 , cot ( − 5 π 6 ) = 3 cos ( − 5 π 6 ) = − 3 2 , sin ( − 5 π 6 ) = − 1 2 , tan ( − 5 π 6 ) = 3 3 , sec ( − 5 π 6 ) = − 2 3 3 , csc ( − 5 π 6 ) = −2 , cot ( − 5 π 6 ) = 3

Use reference angles to find all six trigonometric functions of − 7 π 4 . − 7 π 4 .

## Using Even and Odd Trigonometric Functions

To be able to use our six trigonometric functions freely with both positive and negative angle inputs, we should examine how each function treats a negative input. As it turns out, there is an important difference among the functions in this regard.

Consider the function f ( x ) = x 2 , f ( x ) = x 2 , shown in Figure 5 . The graph of the function is symmetrical about the y -axis. All along the curve, any two points with opposite x -values have the same function value. This matches the result of calculation: ( 4 ) 2 = ( −4 ) 2 , ( −5 ) 2 = ( 5 ) 2 , ( 4 ) 2 = ( −4 ) 2 , ( −5 ) 2 = ( 5 ) 2 , and so on. So f ( x ) = x 2 f ( x ) = x 2 is an even function, a function such that two inputs that are opposites have the same output. That means f ( − x ) = f ( x ) . f ( − x ) = f ( x ) .

Now consider the function f ( x ) = x 3 , f ( x ) = x 3 , shown in Figure 6 . The graph is not symmetrical about the y -axis. All along the graph, any two points with opposite x -values also have opposite y -values. So f ( x ) = x 3 f ( x ) = x 3 is an odd function, one such that two inputs that are opposites have outputs that are also opposites. That means f ( − x ) = − f ( x ) . f ( − x ) = − f ( x ) .

We can test whether a trigonometric function is even or odd by drawing a unit circle with a positive and a negative angle, as in Figure 7 . The sine of the positive angle is y . y . The sine of the negative angle is − y . − y . The sine function, then, is an odd function. We can test each of the six trigonometric functions in this fashion. The results are shown in Table 2 .

## Even and Odd Trigonometric Functions

An even function is one in which f ( − x ) = f ( x ) . f ( − x ) = f ( x ) .

An odd function is one in which f ( − x ) = − f ( x ) . f ( − x ) = − f ( x ) .

Cosine and secant are even:

Sine, tangent, cosecant, and cotangent are odd:

## Using Even and Odd Properties of Trigonometric Functions

If the secant of angle t t is 2, what is the secant of − t ? − t ?

Secant is an even function. The secant of an angle is the same as the secant of its opposite. So if the secant of angle t t is 2, the secant of − t − t is also 2.

If the cotangent of angle t t is 3 , 3 , what is the cotangent of − t ? − t ?

## Recognizing and Using Fundamental Identities

We have explored a number of properties of trigonometric functions. Now, we can take the relationships a step further, and derive some fundamental identities. Identities are statements that are true for all values of the input on which they are defined. Usually, identities can be derived from definitions and relationships we already know. For example, the Pythagorean Identity we learned earlier was derived from the Pythagorean Theorem and the definitions of sine and cosine.

## Fundamental Identities

We can derive some useful identities from the six trigonometric functions. The other four trigonometric functions can be related back to the sine and cosine functions using these basic relationships:

## Using Identities to Evaluate Trigonometric Functions

- ⓐ Given sin ( 45° ) = 2 2 , cos ( 45° ) = 2 2 , sin ( 45° ) = 2 2 , cos ( 45° ) = 2 2 , evaluate tan ( 45° ) . tan ( 45° ) .
- ⓑ Given sin ( 5 π 6 ) = 1 2 , cos ( 5 π 6 ) = − 3 2 , sin ( 5 π 6 ) = 1 2 , cos ( 5 π 6 ) = − 3 2 , evaluate sec ( 5 π 6 ) . sec ( 5 π 6 ) .

Because we know the sine and cosine values for these angles, we can use identities to evaluate the other functions.

- ⓐ tan ( 45° ) = sin ( 45° ) cos ( 45° ) = 2 2 2 2 = 1 tan ( 45° ) = sin ( 45° ) cos ( 45° ) = 2 2 2 2 = 1
- ⓑ sec ( 5 π 6 ) = 1 cos ( 5 π 6 ) = 1 − 3 2 = −2 3 1 = −2 3 = − 2 3 3 sec ( 5 π 6 ) = 1 cos ( 5 π 6 ) = 1 − 3 2 = −2 3 1 = −2 3 = − 2 3 3

Evaluate csc ( 7 π 6 ) . csc ( 7 π 6 ) .

## Using Identities to Simplify Trigonometric Expressions

Simplify sec t tan t . sec t tan t .

We can simplify this by rewriting both functions in terms of sine and cosine.

By showing that sec t tan t sec t tan t can be simplified to csc t , csc t , we have, in fact, established a new identity.

Simplify ( tan t ) ( cos t ) . ( tan t ) ( cos t ) .

## Alternate Forms of the Pythagorean Identity

We can use these fundamental identities to derive alternate forms of the Pythagorean Identity, cos 2 t + sin 2 t = 1. cos 2 t + sin 2 t = 1. One form is obtained by dividing both sides by cos 2 t . cos 2 t .

The other form is obtained by dividing both sides by sin 2 t . sin 2 t .

## Using Identities to Relate Trigonometric Functions

If cos ( t ) = 12 13 cos ( t ) = 12 13 and t t is in quadrant IV, as shown in Figure 8 , find the values of the other five trigonometric functions.

We can find the sine using the Pythagorean Identity, cos 2 t + sin 2 t = 1 , cos 2 t + sin 2 t = 1 , and the remaining functions by relating them to sine and cosine.

The sign of the sine depends on the y -values in the quadrant where the angle is located. Since the angle is in quadrant IV, where the y -values are negative, its sine is negative, − 5 13 . − 5 13 .

The remaining functions can be calculated using identities relating them to sine and cosine.

If sec ( t ) = − 17 8 sec ( t ) = − 17 8 and 0 < t < π , 0 < t < π , find the values of the other five functions.

As we discussed at the beginning of the chapter, a function that repeats its values in regular intervals is known as a periodic function. The trigonometric functions are periodic. For the four trigonometric functions, sine, cosine, cosecant and secant, a revolution of one circle, or 2 π , 2 π , will result in the same outputs for these functions. And for tangent and cotangent, only a half a revolution will result in the same outputs.

Other functions can also be periodic. For example, the lengths of months repeat every four years. If x x represents the length time, measured in years, and f ( x ) f ( x ) represents the number of days in February, then f ( x + 4 ) = f ( x ) . f ( x + 4 ) = f ( x ) . This pattern repeats over and over through time. In other words, every four years, February is guaranteed to have the same number of days as it did 4 years earlier. The positive number 4 is the smallest positive number that satisfies this condition and is called the period. A period is the shortest interval over which a function completes one full cycle—in this example, the period is 4 and represents the time it takes for us to be certain February has the same number of days.

## Period of a Function

The period P P of a repeating function f f is the number representing the interval such that f ( x + P ) = f ( x ) f ( x + P ) = f ( x ) for any value of x . x .

The period of the cosine, sine, secant, and cosecant functions is 2 π . 2 π .

The period of the tangent and cotangent functions is π . π .

## Finding the Values of Trigonometric Functions

Find the values of the six trigonometric functions of angle t t based on Figure 9 .

Find the values of the six trigonometric functions of angle t t based on Figure 10 .

## Finding the Value of Trigonometric Functions

If sin ( t ) = − 3 2 and cos ( t ) = 1 2 , find sec ( t ) , csc ( t ) , tan ( t ) , cot ( t ) . sin ( t ) = − 3 2 and cos ( t ) = 1 2 , find sec ( t ) , csc ( t ) , tan ( t ) , cot ( t ) .

sin ( t ) = 2 2 and cos ( t ) = 2 2 , find sec ( t ) , csc ( t ) , tan ( t ) , and cot ( t ) sin ( t ) = 2 2 and cos ( t ) = 2 2 , find sec ( t ) , csc ( t ) , tan ( t ) , and cot ( t )

## Evaluating Trigonometric Functions with a Calculator

We have learned how to evaluate the six trigonometric functions for the common first-quadrant angles and to use them as reference angles for angles in other quadrants. To evaluate trigonometric functions of other angles, we use a scientific or graphing calculator or computer software. If the calculator has a degree mode and a radian mode, confirm the correct mode is chosen before making a calculation.

Evaluating a tangent function with a scientific calculator as opposed to a graphing calculator or computer algebra system is like evaluating a sine or cosine: Enter the value and press the TAN key. For the reciprocal functions, there may not be any dedicated keys that say CSC, SEC, or COT. In that case, the function must be evaluated as the reciprocal of a sine, cosine, or tangent.

If we need to work with degrees and our calculator or software does not have a degree mode, we can enter the degrees multiplied by the conversion factor π 180 π 180 to convert the degrees to radians. To find the secant of 30° , 30° , we could press

Given an angle measure in radians, use a scientific calculator to find the cosecant.

- If the calculator has degree mode and radian mode, set it to radian mode.
- Enter: 1 / 1 /
- Enter the value of the angle inside parentheses.
- Press the SIN key.
- Press the = key.

Given an angle measure in radians, use a graphing utility/calculator to find the cosecant.

- If the graphing utility has degree mode and radian mode, set it to radian mode.
- Press the ENTER key.

## Evaluating the Cosecant Using Technology

Evaluate the cosecant of 5 π 7 . 5 π 7 .

For a scientific calculator, enter information as follows:

Evaluate the cotangent of − π 8 . − π 8 .

Access these online resources for additional instruction and practice with other trigonometric functions.

- Determing Trig Function Values
- More Examples of Determining Trig Functions
- Pythagorean Identities
- Trig Functions on a Calculator

## 7.4 Section Exercises

On an interval of [ 0 , 2 π ) , [ 0 , 2 π ) , can the sine and cosine values of a radian measure ever be equal? If so, where?

What would you estimate the cosine of π π degrees to be? Explain your reasoning.

For any angle in quadrant II, if you knew the sine of the angle, how could you determine the cosine of the angle?

Describe the secant function.

Tangent and cotangent have a period of π . π . What does this tell us about the output of these functions?

For the following exercises, find the exact value of each expression.

tan π 6 tan π 6

sec π 6 sec π 6

csc π 6 csc π 6

cot π 6 cot π 6

tan π 4 tan π 4

sec π 4 sec π 4

csc π 4 csc π 4

cot π 4 cot π 4

tan π 3 tan π 3

sec π 3 sec π 3

csc π 3 csc π 3

cot π 3 cot π 3

For the following exercises, use reference angles to evaluate the expression.

tan 5 π 6 tan 5 π 6

sec 7 π 6 sec 7 π 6

csc 11 π 6 csc 11 π 6

cot 13 π 6 cot 13 π 6

tan 7 π 4 tan 7 π 4

sec 3 π 4 sec 3 π 4

csc 5 π 4 csc 5 π 4

cot 11 π 4 cot 11 π 4

tan 8 π 3 tan 8 π 3

sec 4 π 3 sec 4 π 3

csc 2 π 3 csc 2 π 3

cot 5 π 3 cot 5 π 3

tan 225° tan 225°

sec 300° sec 300°

csc 150° csc 150°

cot 240° cot 240°

tan 330° tan 330°

sec 120° sec 120°

csc 210° csc 210°

cot 315° cot 315°

If sin t = 3 4 , sin t = 3 4 , and t t is in quadrant II, find cos t , sec t , csc t , tan t , cos t , sec t , csc t , tan t , and cot t . cot t .

If cos t = − 1 3 , cos t = − 1 3 , and t t is in quadrant III, find sin t , sec t , csc t , tan t , sin t , sec t , csc t , tan t , and cot t . cot t .

If tan t = 12 5 tan t = 12 5 , and 0 ≤ t < π 2 0 ≤ t < π 2 , find sin t , cos t , sec t , csc t , and cot t . sin t , cos t , sec t , csc t , and cot t .

If sin t = 3 2 sin t = 3 2 and cos t = 1 2 , cos t = 1 2 , find sec t , csc t , tan t , sec t , csc t , tan t , and cot t . cot t .

If sin 40° ≈ 0.643 sin 40° ≈ 0.643 and cos 40° ≈ 0.766 , cos 40° ≈ 0.766 , find sec 40° , csc 40° , tan 40° , sec 40° , csc 40° , tan 40° , and cot 40° . cot 40° .

If sin t = 2 2 , sin t = 2 2 , what is the sin ( − t ) ? sin ( − t ) ?

If cos t = 1 2 , cos t = 1 2 , what is the cos ( − t ) ? cos ( − t ) ?

If sec t = 3.1 , sec t = 3.1 , what is the sec ( − t ) ? sec ( − t ) ?

If csc t = 0.34 , csc t = 0.34 , what is the csc ( − t ) ? csc ( − t ) ?

If tan t = −1.4 , tan t = −1.4 , what is the tan ( − t ) ? tan ( − t ) ?

If cot t = 9.23 , cot t = 9.23 , what is the cot ( − t ) ? cot ( − t ) ?

For the following exercises, use the angle in the unit circle to find the value of the each of the six trigonometric functions.

For the following exercises, use a graphing calculator to evaluate to three decimal places.

csc 5 π 9 csc 5 π 9

cot 4 π 7 cot 4 π 7

sec π 10 sec π 10

tan 5 π 8 tan 5 π 8

tan 98° tan 98°

cot 33° cot 33°

cot 140° cot 140°

sec 310° sec 310°

For the following exercises, use identities to evaluate the expression.

If tan ( t ) ≈ 2.7 , tan ( t ) ≈ 2.7 , and sin ( t ) ≈ 0.94 , sin ( t ) ≈ 0.94 , find cos ( t ) . cos ( t ) .

If tan ( t ) ≈ 1.3 , tan ( t ) ≈ 1.3 , and cos ( t ) ≈ 0.61 , cos ( t ) ≈ 0.61 , find sin ( t ) . sin ( t ) .

If csc ( t ) ≈ 3.2 , csc ( t ) ≈ 3.2 , and cos ( t ) ≈ 0.95 , cos ( t ) ≈ 0.95 , find tan ( t ) . tan ( t ) .

If cot ( t ) ≈ 0.58 , cot ( t ) ≈ 0.58 , and cos ( t ) ≈ 0.5 , cos ( t ) ≈ 0.5 , find csc ( t ) . csc ( t ) .

Determine whether the function f ( x ) = 2 sin x cos x f ( x ) = 2 sin x cos x is even, odd, or neither.

Determine whether the function f ( x ) = 3 sin 2 x cos x + sec x f ( x ) = 3 sin 2 x cos x + sec x is even, odd, or neither.

Determine whether the function f ( x ) = sin x − 2 cos 2 x f ( x ) = sin x − 2 cos 2 x is even, odd, or neither.

Determine whether the function f ( x ) = csc 2 x + sec x f ( x ) = csc 2 x + sec x is even, odd, or neither.

For the following exercises, use identities to simplify the expression.

csc t tan t csc t tan t

sec t csc t sec t csc t

## Real-World Applications

The amount of sunlight in a certain city can be modeled by the function h = 15 cos ( 1 600 d ) , h = 15 cos ( 1 600 d ) , where h h represents the hours of sunlight, and d d is the day of the year. Use the equation to find how many hours of sunlight there are on February 11, the 42 nd day of the year. State the period of the function.

The amount of sunlight in a certain city can be modeled by the function h = 16 cos ( 1 500 d ) , h = 16 cos ( 1 500 d ) , where h h represents the hours of sunlight, and d d is the day of the year. Use the equation to find how many hours of sunlight there are on September 24, the 267th day of the year. State the period of the function.

The equation P = 20 sin ( 2 π t ) + 100 P = 20 sin ( 2 π t ) + 100 models the blood pressure, P , P , where t t represents time in seconds. (a) Find the blood pressure after 15 seconds. (b) What are the maximum and minimum blood pressures?

The height of a piston, h , h , in inches, can be modeled by the equation y = 3 sin x + 1 , y = 3 sin x + 1 , where x x represents the crank angle. Find the height of the piston when the crank angle is 55° . 55° .

The height of a piston, h , h , in inches, can be modeled by the equation y = 2 cos x + 5 , y = 2 cos x + 5 , where x x represents the crank angle. Find the height of the piston when the crank angle is 55° . 55° .

As an Amazon Associate we earn from qualifying purchases.

Want to cite, share, or modify this book? This book uses the Creative Commons Attribution License and you must attribute OpenStax.

Access for free at https://openstax.org/books/algebra-and-trigonometry-2e/pages/1-introduction-to-prerequisites

- Authors: Jay Abramson
- Publisher/website: OpenStax
- Book title: Algebra and Trigonometry 2e
- Publication date: Dec 21, 2021
- Location: Houston, Texas
- Book URL: https://openstax.org/books/algebra-and-trigonometry-2e/pages/1-introduction-to-prerequisites
- Section URL: https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-4-the-other-trigonometric-functions

© Jun 22, 2023 OpenStax. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution License . The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo are not subject to the Creative Commons license and may not be reproduced without the prior and express written consent of Rice University.

## Trigonometry (Algebra 2 Curriculum - Unit 12) | All Things Algebra®

- Google Apps™

## What educators are saying

Also included in.

## Description

Due to the length of this Trigonometry Unit Bundle , it is divided into two parts with two unit tests. In addition to the unit tests, each part includes guided notes, homework assignments, quizzes, and study guides to cover the following topics:

Unit 12 Part I:

• Pythagorean Theorem

• Special Right Triangles

• Trigonometric Functions (sin, cos, tan, csc, sec, cot)

• Finding Side and Angle Measures

• Applications: Angle of Elevation and Depression

• Angles in Standard Position

• Converting between Degrees and Radians

• Coterminal and Reference Angles

• Trigonometric Functions in the Coordinate Plane

• The Unit Circle

• Law of Sines

• Law of Cosines

• Area of Triangles

• Applications of Law of Sines, Law of Cosines, and Area

Unit 12 Part II:

• Graphing Trigonometric Functions

• Trigonometric Identities

• Sum and Difference of Angle Identities

• Double-Angle and Half-Angle Identities

• Solving Trigonometric Equations

ADDITIONAL COMPONENTS INCLUDED:

(1) Links to Instructional Videos: Links to videos of each lesson in the unit are included. Videos were created by fellow teachers for their students using the guided notes and shared in March 2020 when schools closed with no notice. Please watch through first before sharing with your students. Many teachers still use these in emergency substitute situations. (2) Editable Assessments: Editable versions of each quiz and the unit test are included. PowerPoint is required to edit these files. Individual problems can be changed to create multiple versions of the assessment. The layout of the assessment itself is not editable. If your Equation Editor is incompatible with mine (I use MathType), simply delete my equation and insert your own.

(3) Google Slides Version of the PDF: The second page of the Video links document contains a link to a Google Slides version of the PDF. Each page is set to the background in Google Slides. There are no text boxes; this is the PDF in Google Slides. I am unable to do text boxes at this time but hope this saves you a step if you wish to use it in Slides instead!

This resource is included in the following bundle(s):

Algebra 2 Curriculum

More Algebra 2 Units:

Unit 1 – Equations and Inequalities

Unit 2 – Linear Functions and Systems

Unit 3 – Parent Functions and Transformations

Unit 4 – Solving Quadratics and Complex Numbers

Unit 5 – Polynomial Functions

Unit 6 – Radical Functions

Unit 7 – Exponential and Logarithmic Functions

Unit 8 – Rational Functions

Unit 9 – Conic Sections

Unit 10 – Sequences and Series

Unit 11 – Probability and Statistics

LICENSING TERMS: This purchase includes a license for one teacher only for personal use in their classroom. Licenses are non-transferable , meaning they can not be passed from one teacher to another. No part of this resource is to be shared with colleagues or used by an entire grade level, school, or district without purchasing the proper number of licenses. If you are a coach, principal, or district interested in transferable licenses to accommodate yearly staff changes, please contact me for a quote at [email protected].

COPYRIGHT TERMS: This resource may not be uploaded to the internet in any form, including classroom/personal websites or network drives, unless the site is password protected and can only be accessed by students.

© All Things Algebra (Gina Wilson), 2012-present

## Questions & Answers

All things algebra.

- We're hiring
- Help & FAQ
- Privacy policy
- Student privacy
- Terms of service
- Tell us what you think

Finished Papers

## Leave a Reply Cancel reply

Your email address will not be published. Required fields are marked *

Save my name, email, and website in this browser for the next time I comment.

## What is a good essay writing service?

Oddly enough, but many people still have not come across a quality service. A large number of users fall for deceivers who take their money without doing their job. And some still fulfill the agreements, but very badly.

A good essay writing service should first of all provide guarantees:

- confidentiality of personal information;
- for the terms of work;
- for the timely transfer of the text to the customer;
- for the previously agreed amount of money.

The company must have a polite support service that will competently advise the client, answer all questions and support until the end of the cooperation. Also, the team must get out of conflict situations correctly.

It is necessary to have several payment methods on the site to make it easier for the client to transfer money.

And of course, only highly qualified writers with a philological education should be present in the team, who will not make spelling and punctuation errors in the text, checking all the information and not stealing it from extraneous sites.

- Join for FREE
- Printable Worksheets
- Online Lessons
- Test Maker™
- Printable Games
- Worksheet Generator
- Plans & Pricing

Printable & online resources for educators

- Test Maker TM
- Browse All Questions
- Questions With Images
- Advanced Search

## Share/Like This Page

Filter by grade.

You are browsing Grade 12 questions. View questions in All Grades .

Grade 10 Grade 11 Grade 12 College

## Browse Questions

- All Subjects w/ Images (7037)
- By ELA/Literacy Standard
- By Math Standard
- All Subjects (7606)
- English Language Arts (1766)
- English as a Second Language ESL (785)
- Health and Medicine (449)
- Life Skills (702)

## Arithmetic and Number Concepts

Function and algebra concepts, geometry and measurement, statistics and probability concepts, trigonometry.

- Physical Education (235)
- Science (1777)
- Social Studies (1250)
- Study Skills and Strategies (32)
- Technology (122)
- Vocational Education (197)

## Twelfth Grade (Grade 12) Trigonometry Questions

You can create printable tests and worksheets from these Grade 12 Trigonometry questions! Select one or more questions using the checkboxes above each question. Then click the add selected questions to a test button before moving to another page.

- [math]pi/4[/math]
- [math]pi/5[/math]
- [math]sqrt2/2pi[/math]
- none of the above
- [math]-1[/math]
- [math]sqrt2/2[/math]
- [math]1[/math]
- [math]1/2[/math]
- [math]360deg[/math]
- [math]240deg[/math]
- [math]60deg[/math]
- [math]315deg[/math]
- all of the above
- [math]pi[/math]
- [math]2pi[/math]
- [math]1/2pi[/math]
- [math]4pi[/math]
- [math]sin x[/math]
- [math]cos x[/math]
- [math]tan x[/math]
- [math]sqrt3/2[/math]
- [math]2/sqrt3[/math]
- Privacy Policy
- Terms of Use
- FREE Printable Worksheets
- Common Core ELA Worksheets
- Common Core Math Worksheets

## IMAGES

## VIDEO

## COMMENTS

Chapter 1: Trigonometric Functions Section 1-1: Angles Section 1-2: Angle Relationships and Similar Triangles Section 1-3: Trigonometric Functions Section 1-4: Using the Definitions of the Trigonometric Functions Page 42: Review Exercises Page 45: Test Exercise 1 Exercise 2 Exercise 3 Exercise 4 Exercise 5 Exercise 6 Exercise 7 Exercise 8

7.3 Unit Circle; 7.4 The Other Trigonometric Functions; Chapter Review. Key Terms; Key Equations; Key Concepts; Exercises. Review Exercises; Practice Test; 8 Periodic Functions. ... 12.1 The Ellipse; 12.2 The Hyperbola; 12.3 The Parabola; 12.4 Rotation of Axes; 12.5 Conic Sections in Polar Coordinates; Chapter Review. Key Terms;

Trigonometry Free math problem solver answers your trigonometry homework questions with step-by-step explanations.

1 triangle. Ambiguous Cases: If a>b. 1 right triangle. Ambiguous Cases: If a=x. 0 triangles. Ambiguous Cases: If a<x. 2 triangles. Ambiguous Cases: If a>x. Study with Quizlet and memorize flashcards containing terms like trigonometric ratios, oblique, law of cosines and more.

Algebra and Trigonometry 1e (OpenStax) 7: The Unit Circle - Sine and Cosine Functions 7.3: Unit Circle

Unit 12: Trigonometry. blank_unit_circle.pdf: File Size: 16 kb: File Type: pdf: Download File. completed_unit_circle.pdf: File Size: 246 kb: File Type: pdf: Download File. Notes. 12.1_notes_angles_and_radians.pdf: ... Homework Solutions will now be posted after the homework has been stamped or collected. Please try the problems on your own and ...

Unit 12. Modeling. Course challenge. Test your knowledge of the skills in this course. Start Course challenge. Math; Algebra 2; Unit 11: Trigonometry. 1,700 possible mastery points. ... Trig unit circle review (Opens a modal) Practice. Unit circle Get 3 of 4 questions to level up! Radians. Learn. Intro to radians (Opens a modal) Radians & degrees

Math. Advanced Math. Advanced Math questions and answers. Name: Unit 12 Test Study Guide (Trigonometry - Part 1) Date: Block Topic Trigonometrie Functions 1. Find the values of the six trigonometric functions for angle 1). Give answers in simplest form. cos o . _ 2. If tan find the remaining trigonometric functions. 3.

For the following exercises, use the given information to find the lengths of the other two sides of the right triangle. 22. cos B = 35, a = 6 cos B = 3 5, a = 6. 23. tan A = 5 9, b = 6 tan A = 5 9, b = 6. For the following exercises, use Figure 1 to evaluate each trigonometric function. Figure 1.

7 The Unit Circle: Sine and Cosine Functions. Introduction to The Unit Circle: Sine and Cosine Functions; 7.1 Angles; 7.2 Right Triangle Trigonometry; 7.3 Unit Circle; 7.4 The Other Trigonometric Functions; Chapter Review. Key Terms; Key Equations; Key Concepts; Exercises. Review Exercises; Practice Test ... x = 12 7 x = 12 7. 17. No solution ...

Learn Getting ready for right triangles and trigonometry Hypotenuse, opposite, and adjacent Side ratios in right triangles as a function of the angles Using similarity to estimate ratio between side lengths Using right triangle ratios to approximate angle measure Right triangles & trigonometry: FAQ Practice

5.2 Unit Circle: Sine and Cosine Functions; 5.3 The Other Trigonometric Functions; ... Introduction to Trigonometric Identities and Equations; 7.1 Solving Trigonometric Equations with Identities; 7.2 Sum and Difference Identities; 7.3 Double-Angle, Half-Angle, ... 3 π 12, 5 π 12, 11 π 12, 13 π 12, ...

1.2 Section Exercises. 1. No, the two expressions are not the same. An exponent tells how many times you multiply the base. So 2 3 is the same as 2 × 2 × 2, which is 8. 3 2 is the same as 3 × 3, which is 9. 3. It is a method of writing very small and very large numbers. 5.

The trigonometric functions are periodic. For the four trigonometric functions, sine, cosine, cosecant and secant, a revolution of one circle, or 2 π, will result in the same outputs for these functions. And for tangent and cotangent, only a half a revolution will result in the same outputs.

Be the first to ask All Things Algebra a question about this product. Due to the length of this Trigonometry Unit Bundle, it is divided into two parts with two unit tests. In addition to the unit tests, each part includes guided notes, homework assignments, quizzes, and study guides to cover the following topics:Unit 12 Part I:• Pythagorean ...

Unit 12 Trigonometery. The last unit of the year focuses on right triangle trigonometry. We will learn how to use the sine, cosine and tangent ratios to find missing sides lengths and angle measures in a right triangle. Please see the schedule below. Č.

This video explains exercise 12.7 of chapter 12 and it explains how to find area of a triangle with three different cases.Exercise 12.7 Complete || FSC Math ...

Unit 7 - Trigonometric Functions. Trigonometry is essentially the study of how lengths vary compared to the rotations or angles that create the length. Thus, this unit begins with careful development of essential angle and rotation terminology, including reference angles, quadrant work, and radian angle measurement.

if the sum of the squares of the lengths of two sides of a triangle is equal to the square of the length of the third side, then the triangle is a right triangle. trigonometric ratio. a ratio of the lengths of sides of a right triangle. special right triangles. refers to the 45-45-90 and 30-60-90 right triangles. angle of elevation/depression.

They can complete any type of homework or coursework, regardless of field of study, complexity, and urgency. When you contact the company Essayswriting, the support service immediately explains the terms of cooperation to you. ... Unit 12 Trigonometry Homework 7 Answers: About Writer. 4.9/5. Any paper at any academic level.

Twelfth Grade (Grade 12) Trigonometry Questions. You can create printable tests and worksheets from these Grade 12 Trigonometry questions! Select one or more questions using the checkboxes above each question. Then click the add selected questions to a test button before moving to another page. Convert to radians.