Pre-Calculus
Pre-Calculus Name
Chapter 6 – Graphs of Trigonometric Functions Period
1. Change the degree measures to radian measures in terms of π.
a. 145° b. [pic]
2. Change the radian measures to degree measures.
a. [pic] b. – 3.5
3. Use the unit circle to find the exact values. No Calculator!
a. sin ( – π ) b. cos [pic]
c. sin [pic] d. tan [pic]
e. sec 4π f. csc [pic]
g. cos [pic] h. sin [pic]
i. sin [pic] j. tan [pic]
k. cot [pic] l. sec [pic]
4. The radius of a circle is 9 inches. If a central angle measures [pic], find the length of its intercepted arc.
Formula: [pic] .
5. The diameter of a circle is 24 inches. If a central angle measures 75º, find the length of its intercepted arc.
Formula: [pic] .
6. Find the area of a sector if the central angle measures [pic] radians and the radius of the circle is
10 centimeters.
Formula: [pic] .
7. Find the area of a sector if the central angle measures 82° and the diameter of the circle is
7.3 centimeters. Round answer to 3 decimal places.
8. If the area of a sector is 15 in² and its central angle is 0.2 radians, then find a) the radius of the circle and b) the arc length of the sector. Write answers in exact form.
9. Determine the angular velocity, w , of each situation. Formula: [pic].
a. 1.8 revolutions in 9 seconds b. 250 revolutions in 32 minutes
10. Determine the linear velocity, v , of each situation. Formula: [pic] or [pic].
a. w = 27.4 radians per second , r = 4 feet b. w = 200 revolutions in 160 seconds , r = 18 in
11. The diameter of a Ferris Wheel is 80 feet.
a) If the Ferris Wheel makes one revolution every 45 seconds, then find the linear velocity of a person riding in the Ferris Wheel.
b) Suppose the linear velocity of a person riding the Ferris Wheel is 8 ft/sec. What is the time for one revolution of the Ferris Wheel?
Pre-Calculus Name
Chapter 6 – Graphs of Trigonometric Functions Period
State the a) amplitude, b) the period, and then c) graph each trigonometric function. Sketch at least two periods.
1. [pic]
a) amplitude =
b) period =
2. [pic]
a) amplitude =
b) period =
3. [pic]
a) amplitude =
b) period =
4. [pic]
a) amplitude =
b) period =
5. Find the values on the interval [ 0 , 2π ] for which the following are true.
a) [pic] b) [pic] is undefined
c) [pic] d) [pic]
6. Write an equation of the sine function with the given amplitude and period.
a) amplitude 6, period 10π b) amplitude 0.5, period 6
7. Write an equation of the cosine function with the given amplitude and period.
a) amplitude 4, period [pic] b) amplitude [pic] , period [pic]
Pre-Calculus Name
Chapter 6 – Graphs of Trigonometric Functions Period
State the a) amplitude, b) the period, c) the phase shift, d) the vertical shift, and then e) graph each trigonometric function. Sketch one period from the “start” of the function.
1. [pic]
a) amplitude = c) phase shift =
b) period = d) vertical shift =
2. [pic]
a) amplitude = c) phase shift =
b) period = d) vertical shift =
3. [pic]
a) amplitude = c) phase shift =
b) period = d) vertical shift =
4. [pic]
a) amplitude = c) phase shift =
b) period = d) vertical shift =
5. Write an equation of the sine function with amplitude 2, period π, phase shift [pic] ,
and vertical shift 1.
6. Write an equation of the cosine function with amplitude 6, period [pic], phase shift [pic],
and vertical shift -5.
7. Find the values in the domain of all real numbers for which the following are true.
a) [pic] b) [pic] is undefined
c) [pic] d) [pic]
Pre-Calculus Name
Chapter 6 – Graphs of Trigonometric Functions Period
1. Arnold is a bodybuilder whose weight fluctuates throughout the school year. His weight is given in the table below during the 10 months in which he cycles his training between bulking up, cutting weight, then getting back to his normal weight.
a) Find the amplitude of the sinusoidal function the can
model Arnold’s weight during his 10 month training cycle.
b) Find the vertical shift. c) What is the period?
d) Write a sine function to represent Arnold’s weight as a function of t , time in months, where September is t = 1.
e) According to your model, what is Arnold’s weight half way through his cycle in February? How does this value compare to his actual weight in February?
f) Use your calculator’s Sine Regression to find a sine function to represent Arnold’s weight.
2. The profit of a small business in town can be modeled by a cosine function. A high of $5000 was recorded on January 1st, which reflected the sales generated during the holiday season. On July 1st, a low of $2400 was recorded. Write a cosine function to represent the profit of the business as a function of the time of the year. Let t be the number of months from January 1st , where January 1st is t = 0.
3. Change [pic] to radian measure in terms of (.
4. Change [pic] radians to degree measure.
5. Given a central angle of 76.4(, find the length of the intercepted arc in a circle of radius 6 centimeters.
Formula: [pic] .
6. Find the area of sector if the central angle measures [pic] radians and the radius of the circle is 2.6 meters.
Formula: [pic] .
Formula: [pic]. Formula: [pic] or [pic].
7. A belt runs a pulley that has a diameter of 12 centimeters. If the pulley rotates at 80 revolutions per minute, what is its angular velocity in radians per second and its linear velocity in centimeters per second?
8. Use the unit circle to evaluate each expression.
a) tan [pic]
b) cos [pic]
9. Graph [pic] .
a) amplitude =
b) period =
10. State the amplitude and period for each function.
a) [pic] b) [pic]
11. Write an equation of the sine function with amplitude 5 and period [pic] .
Pre-Calculus Name
Chapter 6 – Graphs of Trigonometric Functions Period
1. Write an equation of the tangent function with period 2π, phase shift π, and vertical shift –1.
2. Write an equation of the secant function with period π, phase shift [pic], and vertical shift 2.
3. State the period and phase shift of the function [pic]
State the a) amplitude, b) the period, c) the phase shift, d) the vertical shift, and then e) graph each trigonometric function. Sketch at least two periods of the function.
4. [pic]
a) amplitude = c) phase shift =
b) period = d) vertical shift =
5. [pic]
a) amplitude = c) phase shift =
b) period = d) vertical shift =
Pre-Calculus Name
Chapter 6 – Graphs of Trigonometric Functions Period
1. State the domain and range of the following functions.
a) y = [pic] b) y = Arcsin x c) y = arctan x
D: D: D:
R: R: R:
2. Find the exact value.
a. [pic] b. [pic]
c. [pic] d. [pic]
e. [pic] f. [pic]
State the a) amplitude, b) the period, c) the phase shift, d) the vertical shift, and then e) graph each trigonometric function. Sketch at least two periods.
3. [pic]
a) amplitude = c) phase shift =
b) period = d) vertical shift =
4. [pic]
a) amplitude = c) phase shift =
b) period = d) vertical shift =
5. Find the values of [pic] for which each equation is true.
a) [pic] b) [pic]
-----------------------
6.1 and 6.2 Supplementary Worksheet
6.1
6.2
6.3 and 6.4 Supplementary Worksheet
6.5 Supplementary Worksheet
6.6 Supplementary Worksheet
Sept |Oct |Nov |Dec |Jan |Feb |Mar |Apr |May |June |July | |179 |186 |192 |193 |189 |180 |173 |170 |169 |174 |181 | |
Review: 6.1 to 6.4 QUIZ
6.7 Supplementary Worksheet
6.8 Supplementary Worksheet
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