Precalculus by Richard Wright

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Have I not commanded you? Be strong and courageous. Do not be afraid; do not be discouraged, for the Lord your God will be with you wherever you go. Joshua‬ ‭1‬:‭9‬ ‭NIV‬‬‬‬‬‬‬‬‬‬‬‬

4-02 Unit Circle

Mr. Wright teaches the lesson.

Summary: In this section, you will:

SDA NAD Content Standards (2018): PC.5.3

Carousel at Glen Echo Park, Maryland.
Figure 1: Carousel at Glen Echo Park, Maryland. credit (Flickr/chrisbb)

A carousel takes a child for a ride on a colorful animal in a circle. Using trigonometric functions and the unit circle, we can calculate the location of the rider at any given time.

Unit Circle

Think of a circle centered at the origin with a radius of one. Since the radius is one, it is called a unit circle. A similar circle was used to show angle measures. Now coordinates are added to the circle.

Unit Circle

The Unit Circle is a circle centered at the origin with radius equal to 1. It can be used to evaluate trigonometric functions.

Figure 2: The Unit Circle

Think of a number line wrapped around the circle. This would measure the arc length. Remember arc length is s = and since r = 1, then s = θ. The x and y-coordinates then would be functions of the arc length which equals the angle in radians. These functions are called trigonometric functions.

Trigonometric Functions: Unit Circle
Name Formula Name Formula
Sine sin θ = y Cosecant csc θ = 1y
Cosine cos θ = x Secant sec θ = 1x
Tangent tan θ = yx Cotangent cot θ = xy

Note: These are undefined when the denominator of the fraction equals 0.

Example 1: Evaluate Trigonometric Functions

Evaluate the six trigonometric functions for the given angles.

  1. θ = π
  2. θ=π4
  3. θ=4π3
  4. θ=11π6
Solution
  1. Use the angle on the unit circle to find the corresponding x and y-coordinates. For π, x = −1 and y = 0.

    sin π = y = 0 csc π = 1 y = 1 0 = undefined
    cos π = x = - 1 sec π = 1 x = 1 - 1 = - 1
    tan π = y x = 0 - 1 = 0 cot π = x y = - 1 0 = undefined
  2. Use the angle on the unit circle to find the corresponding x and y-coordinates. For π4, x = 22 and y = 22.

    sin π 4 = y = 2 2 csc π 4 = 1 y = 1 2 2 = 2
    cos π 4 = x = 2 2 sec π 4 = 1 x = 1 2 2 = 2
    tan π 4 = y x = 2 2 2 2 = 1 cot π 4 = x y = 2 2 2 2 = 1
  3. Use the angle on the unit circle to find the corresponding x and y-coordinates. For 4π3, x = -12 and y = -32.

    sin 4 π 3 = y = - 3 2 csc 4 π 3 = 1 y = 1 - 3 2 = - 2 3 3
    cos 4 π 3 = x = - 1 2 sec 4 π 3 = 1 x = 1 - 1 2 = - 2
    tan 4 π 3 = y x = - 3 2 - 1 2 = 3 cot 4 π 3 = x y = - 1 2 - 3 2 = 3 3
  4. Use the angle on the unit circle to find the corresponding x and y-coordinates. For 11π6, x = 32 and y = -12.

    sin 11 π 6 = y = - 1 2 csc 11 π 6 = 1 y = 1 - 1 2 = - 2
    cos 11 π 6 = x = 3 2 sec 11 π 6 = 1 x = 1 3 2 = 2 3 3
    tan 11 π 6 = y x = - 1 2 3 2 = - 3 3 cot 11 π 6 = x y = 3 2 - 1 2 = - 3
Try It 1

Evaluate the six trigonometric functions for θ = π 2 .

Answers

sin π 2 = 1 ; cos π 2 = 0 ; tan π 2 = undefined ; csc π 2 = 1 ; sec π 2 = undefined ; cot π 2 = 0

Example 2: Evaluate Trigonometric Functions Not Between 0 and 2π

Evaluate the six trigonometric functions for the given angles.

a. θ = - π 3 , b. θ = 9 π 4

Solution
  1. θ = - π 3 is not on the unit circle because it is negative. The first step is to find a coterminal angle between 0 and 2π.

    θ = - π 3 + 2 π θ = - π 3 + 6 π 3 θ = 5 π 3

    The coordinates for θ = 5 π 3 are 1 2 - 3 2 . Use those in the trigonometric functions and evaluate.

    sin - π 3 = y = - 3 2 csc - π 3 = 1 y = 1 - 3 2 = - 2 3 3
    cos - π 3 = x = 1 2 sec - π 3 = 1 x = 1 1 2 = 2
    tan - π 3 = y x = - 3 2 1 2 = - 3 cot - π 3 = x y = 1 2 - 3 2 = - 3 3
  2. θ = 9 π 4 is not on the unit circle because it is greater than 2π. The first step is to find a coterminal angle between 0 and 2π.

    θ = 9 π 4 - 2 π θ = 9 π 4 - 8 π 4 θ = π 4

    The coordinates for θ = π 4 are 2 2 2 2 . Use those in the trigonometric functions and evaluate.

    sin 9 π 4 = y = 2 2 csc 9 π 4 = 1 y = 1 2 2 = 2
    cos 9 π 4 = x = 2 2 sec 9 π 4 = 1 x = 1 2 2 = 2
    tan 9 π 4 = y x = 2 2 2 2 = 1 cot 9 π 4 = x y = 2 2 2 2 = 1
Try It 2

Evaluate the six trigonometric functions for θ = 3 π .

Answers

sin 3 π = 0 ; cos 3 π = - 1 ; tan 3 π = 0 ; csc 3 π = undefined ; sec 3 π = - 1 ; cot 3 π = undefined

Even and Odd Trigonometric Identities

Notice the coordinates on the unit circle form a pattern. Moving up or down from the x-axis, the coordinates are the same, only with different signs. This indicates symmetry.

Figure 3: The Unit Circle

Functions symmetric over the y-axis are classified as even. Functions symmetric about the origin are classified as odd. The following formulas are called identities because both sides are equal for all values of the variable.

Even and Odd Trigonometric Identities

Even functions are cosine and secant.

cos(−u) = cos(u) sec(−u) = sec(u)

Odd functions are sine, cosecant, tangent, and cotangent.

sin(−u) = −sin(u) csc(−u) = −csc(u)
tan(−u) = −tan(u) cot(−u) = −cot(u)

Example 3: Even and Odd Trigonometric Functions

If sin(x) = 1, what is sin(−x)?

Solution

Sine is an odd trigonometric function, so use the identity to solve the problem.

sin(−x) = −sin(x)
sin(−x) = −1

Try It 3

If sec(x) = 2, what is −sec(x)?

Answer

2

Use a Calculator

Calculators have two or three units for angles. These only matter when using the trigonometric functions. Make sure your calculator is using the correct angle unit for each problem. The settings are in MODE, Settings, or SETUP.

Example 4: Using a Calculator to Evaluate Trigonometric Functions

Use a calculator to evaluate

  1. sin π 10
  2. tan 2 π 3
  3. sec 120 °
Solution
  1. The angle \(\frac{π}{10}\) is in radians. Make sure the calculator is in radian mode. Then use the calculator.

    On a TI-84 calculator.

    SIN 2ND π ÷ 10 ) ENTER
    sin(π/10) = 0.3090

  2. On a NumWorks calculator.

    sin π ÷ 10 EXE
    \(\sin\left(\frac{π}{10}\right) = 0.3090\)

  3. The angle 2π3 is in radians. Make sure the calculator is in radian mode. Then use the calculator.

    On a TI-84 calculator.

    TAN 2 2ND π ÷ 3 ) ENTER
    tan(2π/3) = −1.7321

    On a NumWorks calculator.

    tan 2 π ÷ 3 EXE
    \(\tan\left(\frac{2π}{3}\right) = −1.7321\)

  4. The angle 120° is in radians. Make sure the calculator is in degree mode. There is no SEC button on most calculators, but secant is the reciprocal of cosine as seen in their formulas.

    sec θ = 1 x cos θ = x Substitute sec θ = 1 cos θ

    Then use the calculator.

    On a TI-84 calculator.

    1 ÷ COS 120 ) ENTER
    1/cos(120) = −2

    On a NumWorks calculator.

    1 ÷ cos 120 EXE
    \(\frac{1}{\cos(120)} = -2\)

Try It 4

Evaluate a. tan 68° and b. cot3π11 using a calculator.

Answer

2.4751, 0.8665

Lesson Summary

Unit Circle

The Unit Circle is a circle centered at the origin with radius equal to 1. It can be used to evaluate trigonometric functions.

The Unit Circle

Trigonometric Functions: Unit Circle
Name Formula Name Formula
Sine sin θ = y Cosecant csc θ = 1y
Cosine cos θ = x Secant sec θ = 1x
Tangent tan θ = yx Cotangent cot θ = xy

Note: These are undefined when the denominator of the fraction equals 0.


Even and Odd Trigonometric Identities

Even functions are cosine and secant.

cos(−u) = cos(u) sec(−u) = sec(u)

Odd functions are sine, cosecant, tangent, and cotangent.

sin(−u) = −sin(u) csc(−u) = −csc(u)
tan(−u) = −tan(u) cot(−u) = −cot(u)

Helpful videos about this lesson.

Practice Exercises

  1. Draw and label the complete unit circle.
  2. Evaluate the six trigonometric functions using the point on the unit circle.

  3. Evaluate all six trigonometric functions for the given angle using the unit circle.

  4. 150°
  5. 7 π 6
  6. 0
  7. - 3 π 2
  8. 480°
  9. - 11 π 4
  10. If tan(x) = 1.5, what is tan(−x)?
  11. If −sec(x) = 2, what is sec(−x)?
  12. Use a calculator to evaluate the expression

  13. cos 5 π 12
  14. sin 100°
  15. csc π 5
  16. Problem Solving

  17. If a child riding a pink horse starts a ride on a carousel at the point (1, 0) and it rotates in a circle around the origin, what is the coordinates of the child after 45 seconds given the carousel rotates at 1 revolution per minute?
  18. Mixed Review

  19. (4-01) a) draw the angle in standard position, b) convert it to the other angle unit, c) find a positive coterminal angle and d) find a negative coterminal angle of 6π7.
  20. (4-01) A race car with an 18-inch diameter wheel is traveling at 180 mi/h. Find the angular speed of the wheels in rad/min. How many revolutions per minute do the wheels make?
  21. (3-02) Evaluate without using a calculator: log3 81.
  22. (3-04) Solve 2 log3 (x − 1) = 10.
  23. (2-08) Identify the asymptotes and graph f x = 2 x - 1 x 2 .

Answers

  1. sinθ=-45 cscθ=-54
    cosθ=35 secθ=53
    tanθ=-43 cotθ=-34
  2. sinθ=725 cscθ=257
    cosθ=-2425 secθ=-2524
    tanθ=-724 cotθ=-247
  3. sinθ=12 cscθ=2
    cosθ=-32 secθ=-233
    tanθ=-33 cotθ=-3
  4. sinθ=-12 cscθ=-2
    cosθ=-32 secθ=-233
    tanθ=33 cotθ=3
  5. sinθ=0 cscθ=undefined
    cosθ=1 secθ=1
    tanθ=0 cotθ=undefined
  6. sinθ=1 cscθ=1
    cosθ=0 secθ=undefined
    tanθ=undefined cotθ=0
  7. sinθ=32 cscθ=233
    cosθ=-12 secθ=-2
    tanθ=-3 cotθ=-33
  8. sinθ=-22 cscθ=-2
    cosθ=-22 secθ=-2
    tanθ=1 cotθ=1
  9. −1.5
  10. −2
  11. 0.2588
  12. 0.9848
  13. 1.7013
  14. (0, −1)
  15. ; 10807°; 20π7; -8π7
  16. 21120 rad/min; 3361 rev/min
  17. 4
  18. 244
  19. VA: x = 0; HA: y = 0;