Download Unit 8.5 Problem Set Name: Period: ______ 1. One of the most

Unit 8.5 Problem Set
Name: _________________________________ Period: ____________
1. One of the most popular amusement park rides is the Ferris wheel. One Ferris wheel has a diameter of 50 feet.
Riders board the cars at ground level and the wheel moves counterclockwise. Each ride consists of four revolutions
and you can assume that the Ferris wheel rotates at a constant rate.
Create a sketch to model the height of a rider above ground with respect to the number of revolutions of the Ferris
wheel. Include 4 revolutions. Then complete the table to represent the height of a rider above ground as a function of
number of revolutions of the Ferris wheel.
2. At a different amusement park, a Ferris wheel was designed so that half of the hweel is actually below ground.
The diameter of this underground Ferris hweel is still 50 feet. The top of the ride reaches 25 feet above ground and
the bottom of the rie reaches 25 feet below ground. Riders board the cars at ground level to the right and the Ferris
wheel moves counterclockwise.
Create a sketch to model the height of a rider above ground with respect to the number of revolutions of the Ferris
wheel. Include 4 revolutions. Then complete the table to represent the height of a rider above ground as a function of
number of revolutions of the Ferris wheel.
3. In your own words, compare and contrast the graphs in problems #1 and #2. (HINT: Are they both periodic
functions? What are the periods and the amplitudes?)
4. In problems #1 and #2, you modeled the height of a rider above ground as a function of the number of revolutions
of two different Ferris wheels. You can also model the height of a rider as a function using angle measures. Use the
following instructions to complete the graph on the next page.
5. Use the following unit circle instead of a protractor, and your knowledge of special right triangles to complete the
graphs for y  sin( x) and y  cos( x) .
y  sin( x)
y  cos( x)
6. Find the period and amplitude of each trigonometric
function.
(A) 𝑦 = 3 sin 2𝑥
(B) 𝑦 = 2 cos 3𝑥
Period:
Period:
Amplitude:
(D) 𝑦 = −3 sin
Period:
𝑥
3
Amplitude:
(G) 𝑦 = −2 sin 𝑥
Period:
Amplitude:
Amplitude:
2
(C) 𝑦 =
5
2
cos
Period:
(E) 𝑦 = 3 sin 𝜋𝑥
(F) 𝑦 =
Period:
Amplitude:
Period:
(H) 𝑦 = − cos
2𝑥
5
(I)
Period:
Amplitude:
Period:
𝑥
2
Amplitude:
3
2
sin
𝜋𝑥
2
Amplitude:
1
𝑦 = 3 cos 4𝜋𝑥
Amplitude:
7. Describe the relationship between the graphs of f and g. Consider amplitude, periods, and translations (shifts).
(A)
(B)
g
f
g
f
(C)
(D)
g
f
f
g
8. Sketch the graph. (Include two full periods and make correct markings on the axes.)
𝑔(𝑥) = −4 sin 𝑥
(B)
(C) 𝑔(𝑥) = 4 + cos 𝑥
(D)
(A)
𝑔(𝑥) = sin
𝑥
3
𝜋
𝑔(𝑥) = cos (𝑥 − 2 )
9. Sketch the graph of the function by hand. Use a graphing utility to verify your sketch. (Include two full periods
and make correct markings on the axes.)
(A)
𝑦 = sin 4𝑥
(B)
𝑦 = cos
𝑥
2
10. Find a and d for the function 𝑓(𝑥) = 𝑎 cos 𝑥 + 𝑑 or 𝑓(𝑥) = 𝑎 sin 𝑥 + 𝑑 such that the graph of f matches the
figure.
(A)
(B)
2
2
2
3
4
4
8
3
11. Determine which function is represented by the graph. Do not use a calculator.
(A)
3
5
(B)
2
2
3
3
3
5
𝑥
𝑦 = 4 cos 4𝑥
(a) 𝑦 = 2 sin 2𝑥
(b) 𝑦 = −2 sin 2
(a)
(c) 𝑦 = −2 sin(−2𝑥)
(d) 𝑦 = −2 cos 2𝑥
(c) 𝑦 = −4 cos(𝑥 + 𝜋)
(b) 𝑦 = 4 cos(𝑥 + 𝜋)
(d)
𝑦 = −4 cos 2𝑥
(C)
(D)
3
4
2
2
4
2
2
(a) 𝑦 = 1 + sin
(c) 𝑦 = 1 − sin
𝜋
2
𝜋
2
1
(b) 𝑦 = 1 + cos
𝜋
2
(d) 𝑦 = 1 − sin 2𝑥
(a) 𝑦 = cos 2𝑥
(c) 𝑦 = cos
𝜋
2
(b) 𝑦 = − cos 2𝑥
(d) 𝑦 = − cos
𝜋
2
12. A company that produces snowboards, which are seasonal products, forecasts monthly sales for 1 year to be: 𝑆 =
𝜋𝑡
74.50 + 43.75 cos
where S is the sales in thousands of units and t is the time in months, with 𝑡 = 1
6
corresponding to January. Use a graphing calculator to graph the sales function over the one year period.
Use the graph to determine the months of maximum and minimum sales.
REVIEW:
13. (A) Sketch the angle in standard position, (B)
determine the quadrant in which the angle lies, and (C)
list one positive and one negative co-terminal angle.
(list your answers in radians – in terms of 𝜋)
14. (A) Sketch the angle in standard position, (B)
determine the quadrant in which the angle lies, and (C)
list one positive and one negative co-terminal angle.
(list your answers in degrees)
210 ∘
7𝜋
− 4
15. Convert the angle measure from degrees to
radians. Round your answer to three decimal places.
16. Convert the angle measure from radians to
degrees.
−355 ∘
3𝜋
− 5
17. Evaluate (without a calculator) the sine, cosine, and tangent of the angle (𝜃).
(A)
𝜃 = −𝜋
(B)
𝜃=
−
5𝜋
6
18. Use a calculator to evaluate the expression. Round your answer to four decimal places.
(A) cot 2.3
(B) sec 4.5
5𝜋
(C) cos 3
20. The point (3, −4) is on the terminal side of an
angle in standard position. Determine the exact values
of the six trigonometric functions of the angle. (hint:
Draw a triangle).
19. Find the exact values of the six trigonometric
functions of the angle 𝜃 shown in the figure.
10
2

21. Evaluate the sine, cosine, and tangent of each angle without using a calculator.
(A)
−
𝜋
6
11𝜋
(D) tan (− 6 )
(B)
10𝜋
3