Pre-Calculus – Trigonometric Functions ~1~ NJCTL.org PRECALCULUS UNIT 4: TRIGONOMETRY PRACTICE PROBLEMS Angle and Radian Measures Convert each degree measure into radians. Round answers to the 4 th decimal place. 1. 34.375° 2. 176.48° 3. 225.8525° Convert each radian measure into degrees. Round answers to the 4 th decimal place. 4. 0.25 radians 5. 1.34 radians 6. 4.28 radians Find the length of each arc. 7. = 235°, = 9 8. = 5.19 , = 7.7 9. = 2.85 , = 11 Given the arc length and central angle , find the radius of the circle. 10. = 298°, = 34 11. = 3 , = 4.9 12. = 3.1 , = 11.7 Convert each degree measure into radians. Round answers to the 4 th decimal place. 13. 14.85° 14. 157.3535° 15. 290.725° Convert each radian measure into degrees. Round answers to the 4 th decimal place. 16. 0.72 radians 17. 2.46 radians 18. 5.11 radians Find the length of each arc. 19. = 135°, = 12 20. = 4.6 , = 3.7 21. = 2.9 , = 39 Given the arc length and central angle , find the radius of the circle. 22. = 198°, = 39 23. = 2 , = 9 24. = 5.1 , = 19.8 Right Triangle Trigonometry & the Unit Circle Find the exact value of the given expression. 25. cos 4 3 26. sin 7 4 27. sec 2 3 28. tan −5 6 29. cot 15 4 30. csc −9 2
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PRECALCULUS UNIT 4: TRIGONOMETRY PRACTICE PROBLEMS Angle and Radian Measures Convert each degree measure into radians. Round answers to the 4th decimal place.
1. 34.375°
2. 176.48°
3. 225.8525°
Convert each radian measure into degrees. Round answers to the 4th decimal place.
4. 0.25 radians
5. 1.34 radians
6. 4.28 radians
Find the length of each arc.
7. 𝜃 = 235°, 𝑟 = 9 𝑐𝑚
8. 𝜃 = 5.19 𝑟𝑎𝑑, 𝑑 = 7.7 𝑚
9. 𝜃 = 2.85 𝑟𝑎𝑑, 𝑟 = 11 𝑚
Given the arc length and central angle 𝜃, find the radius of the circle.
10. 𝜃 = 298°, 𝑠 = 34 𝑚𝑚
11. 𝜃 = 3 𝑟𝑎𝑑, 𝑠 = 4.9 𝑚
12. 𝜃 = 3.1 𝑟𝑎𝑑, 𝑠 = 11.7 𝑐𝑚
Convert each degree measure into radians. Round answers to the 4th decimal place. 13. 14.85°
14. 157.3535°
15. 290.725°
Convert each radian measure into degrees. Round answers to the 4th decimal place.
16. 0.72 radians
17. 2.46 radians
18. 5.11 radians
Find the length of each arc.
19. 𝜃 = 135°, 𝑟 = 12 𝑐𝑚
20. 𝜃 = 4.6 𝑟𝑎𝑑, 𝑑 = 3.7 𝑚
21. 𝜃 = 2.9 𝑟𝑎𝑑, 𝑟 = 39 𝑚𝑚
Given the arc length and central angle 𝜃, find the radius of the circle.
22. 𝜃 = 198°, 𝑠 = 39 𝑐𝑚
23. 𝜃 = 2 𝑟𝑎𝑑, 𝑠 = 9 𝑚
24. 𝜃 = 5.1 𝑟𝑎𝑑, 𝑠 = 19.8 𝑐𝑚
Right Triangle Trigonometry & the Unit Circle Find the exact value of the given expression.
The sine (red; to the left) and cosine (blue; to the right) waves are shown in the graphs below.
Determine if each statement provided is True or False.
55. sin𝜋
4= sin
7𝜋
4
True
False
56. cos𝜋
3= cos
5𝜋
3
True
False
57. sin2 3𝜋
2= 0
True
False
58. cos𝜋
3< cos
7𝜋
4
True
False
59. If sin 𝑘 = 0.866 and cos 𝑘 < 0, what is the exact value of sin(𝑘 + 4𝜋)?
a. 0.866
b. –0.866
c. 0.5
d. –0.5
60. If cos 𝑘 = −0.707 and sin 𝑘 < 0, what is the exact value of cos(𝑘 − 6𝜋)?
a. 0.707
b. –0.707
c. 0.5
d. –0.5
State the amplitude, period, and transformations that occur when comparing it to the graph of its parent trigonometric function (e.g. compare 𝑦 = 2 sin 𝑥 to 𝑦 = sin 𝑥). Draw the graph by hand and then check it with a graphing calculator.
100. The graphs of tangent and cotangent are shown for one cycle, from 0 and 𝜋. Identify
the false statement.
a. cot (2𝜋
3) > tan (
2𝜋
3)
b. cot (𝜋
6) + cot (
5𝜋
6) = 0
c. cot (𝜋
3) > tan (
𝜋
3)
d. If cot 𝑥 = tan 𝑥 = −1, then 𝑥 =3𝜋
4
Graphs of Composite Trigonometric Functions
State the amplitude, period, and transformations that occur when comparing the new function to the graph of its parent trigonometric function (e.g. compare 𝑦 = 2 tan 𝑥 to 𝑦 = tan 𝑥). Draw the graph by hand and then check it with a graphing calculator.
State the amplitude, period, and transformations that occur when comparing the new function to the graph of its parent trigonometric function (e.g. compare 𝑦 = 2 tan 𝑥 to 𝑦 = tan 𝑥). Draw the graph by hand and then check it with a graphing calculator.
133. The table gives the normal daily temperatures in Philadelphia P in degrees Fahrenheit
for month t with t = 1 corresponding to January
The model for these temperatures is given by
𝑃(𝑡) = 63.17 + 22.75 sin (𝜋𝑡
6+ 4.21).
a. Use a graphing utility to graph the data points & the model for the temperatures in Philadelphia. How well does the model fit?
b. Find the average annual temperature in Philadelphia. Which term of the equation is closely related to your average?
c. What is the period in this model? What does it stand for?
134. A mass suspended from a spring is compressed a distance of 4 cm above its rest
position, as shown in the figure. The mass is released after time t = 0 and allowed to
oscillate. It is observed that the mass reaches its lowest point 1
2 second after it is
released. Find an equation that describes the motion of the mass.
135. A Ferris wheel has a radius of 9 m, and the bottom of the wheel passes 1 m above the
ground. The Ferris wheel makes one complete revolution every 20 s, and a person
riding the Ferris wheel is at a minimum value when 𝑡 = 0. Find an equation that gives the height above the ground of a person on the Ferris wheel as a function of time.
136. The table gives the normal daily temperatures in Honolulu H in degrees Fahrenheit for
month t with t = 1 corresponding to January
The model for these temperatures is given by
𝐻(𝑡) = 84.40 + 4.28 sin (𝜋𝑡
6+ 3.86).
a. Use a graphing utility to graph the data points & the model for the temperatures in Honolulu. How well does the model fit?
b. Find the average annual temperature in Honolulu. Which term of the equation is closely related to your average?
c. What is the period in this model? What does it stand for?
137. A mass suspended from a spring is pulled down a distance of 0.6 m from it’s rest
position, as shown in the figure. The mass is released after time t = 0 and allowed to
oscillate. If the mass returns to its rest position after 1 second, find an equation that
describes its motion.
138. A Ferris wheel has a radius of 11 m, and the bottom of the wheel passes 1 m above
the ground. The Ferris wheel makes one complete revolution every 30 s, and a person
riding the Ferris wheel is at a minimum value when 𝑡 = 0. Find an equation that gives the height above the ground of a person on the Ferris wheel as a function of time.