Trig Exam 2 Review F07 O’Brien
Trigonometry Exam 2 Review: Chapters 4, 5, 6
25 – 30% of the questions on Exam 2 will come from Chapters 1 through 3. The other 70 – 75% of the exam will come from Chapters 4 through 6. There may be some fill-in-the blank, matching, true-false, and/or multiple choice questions, as well as
problems you must work out.
To prepare for the second exam, I’d suggest you do the following:
1.Go over your notes and the Quick Reviews at the end ofevery chapter from 1 through 6.
2.Go over the Concept Questions in each section from 1.1 through 6.3.
3.Rework Exam 1, and work the problems on this review,using only thedepartmental formula sheet. Try not to peek at
your notes, homework, text, solutions manual, or other resources.
4.Try to finish the review a few days before the exam so you have time to go back through it and make sure you can do
all the problems on your own.
Directions:On every problem, show all of your support work and / or explain how you came up with your answer.
Anytime you are asked to perform a calculation manually or to give an exact answer, you may not use a
calculator.
1.Given
a.Find: a = ______b = ______c = ______d = ______
b.Find: amp: ______x-axis reflection: ______period: ______
x-increment: ______phase shift: ______vertical translation: ______
c.Find: 5 key points:______
d.Manually graph two periods of the given function.
2.Given
a.Find: a = ______b = ______c = ______d = ______
b.Find: amp: ______x-axis reflection: ______period: ______
x-increment: ______phase shift: ______vertical translation: ______
c.Find: 5 key points:______
d.Manually graph two periods of the given function.
3.Given
a.Find: a = ______b = ______c = ______d = ______
b.Find: x-axis reflection: ______period: ______
x-increment: ______phase shift: ______vertical translation: ______
c.Find: left asymptote: x = ______right asymptote: x = ______
d.Find: left key point: ______middle key point: ______right key point: ______
e.Manually graph two periods of the given function.
4.Given
a.Find: a = ______b = ______c = ______d = ______
b.Find: x-axis reflection: ______period: ______
x-increment: ______phase shift: ______vertical translation: ______
c.Find: left asymptote: x = ______right asymptote: x = ______
d.Find: left key point: ______middle key point: ______right key point: ______
e.Manually graph two periods of the given function.
5.Given
a.Find: a = ______b = ______c = ______d = ______
b.Find: amp of sine: ______x-axis reflection: ______period: ______
x-increment: ______phase shift: ______vertical translation: ______
c.Find: 5 key points of sine______
d.Manually graph two periods of the given function.
6.Given
a.Find: a = ______b = ______c = ______d = ______
b.Find: amp of cosine: ______x-axis reflection: ______period: ______
x-increment: ______phase shift: ______vertical translation: ______
c.Find: 5 key points of cosine______
d.Manually graph two periods of the given function.
7.An object is attached to a coiled spring. It is pulled down a distance of 6 units from its equilibrium position and then
released. The time for one complete oscillation is 4 sec.
a.Write an equation that models the position of the object at time t.
b.Determine the position at t = 1.25 sec.
c.Find the frequency.
8.The height attained by a weight attached to a spring set in motion is s(t) = –4 cos 8πt inches after t seconds.
a.Find the maximum height that the weight rises above the equilibrium position of y = 0.
b.When does the weight first reach its maximum height, if t ≥ 0?
c.What are the frequency and period?
9.Given cos s = and tan s < 0, find sin s.
10.Given , find the remaining five trigonometric functions of θ.
11.Use identities to write sec x in terms of sin x.
12.Writein terms of sine and cosine, and simplify so that no quotients appear in the final expression.
13.Verify the identity .
14.Verify the identity.
15.Verify the identity tan θ + cot θ = sec θ csc θ.
16.Verify the identity.
17.Verify the identity.
18.Verify the identity .
19.Use the sum and difference identities to find the exact values of the cosine of 195°. Do not use a calculator.
20.Use the sum & difference identities to write the following expressions as the sine, cosine, or tangent of a single angle.
a. sin 25° cos 15° + cos 25° sin 15° b.
21.Find an angle that makes .
22.Given , in Quadrant IV, and , in Quadrant III, find
a. b. c.
23.Write each expression in terms of a single trigonometric function:
a. 2 sin 3y cos 3y b. c.
24.Use the half-angle identities to find the exact values of the sine, cosine, and tangent of .
25.Given with in Quadrant IV, find the sine, cosine, and tangent of .
26.Given with in Quadrant III, find the sine, cosine, and tangent of .
27.Write cos 7x cos 3x as the sum or difference of two functions.
28.Find the exact value of cos 157.5° sin 22.5°. Do not use a calculator.
29.Write as the product of two functions.
30.Evaluate . Do not use a calculator.
31.Find the exact degree value of . Do not use a calculator.
32.Use a calculator to find the degree measure of θ = arccot (–.3451).
33.Find the exact radian value of. Do not use a calculator.
34.Give the exact value of sec (arcsin .2). Do not use a calculator.
35.Find the exact value of the given expressions. Do not use a calculator except to get a final answer on b.
a. b.
36.Find the exact solutions of 2 sin x – 1 = csc x in the interval [0°, 360°).
37.Find the exact solutions of in the interval [0, 2π).
38.Find the exact solutions of in the interval [0°, 360°).
39.Find the exact solutions of tan x + = sec x in the interval [0, 2π).
40.Find the exactsolutions of in the interval [0, 2π).
41.Find the exactsolutions of in the interval [0, 2π).
42.Find the exactsolutions of in the interval [0, 2π).
Answers
1.a. a = –3, b = 2, c = π, d = 1; b. amp: 3, x-axis ref: yes; period: π,
x-inc: , p.s.: right, v.t.: 1 up
c. ; d.
2.a. a = , b = 1, c = , d = –2; b. amp: , x-axis ref: no;
period: 2π, x-inc: , p.s.: left, v.t.: 2 down
c. ; d.
3.a. a = 2, b = 1, c = , d = 3; b. x-axis ref: no; period: π,
x-inc: , p.s.: right, v.t.: 3 up; c. LA: , RA: ;
d. lkp: (0, 1), mkp: , rkp: ; e.
4.a. a = , b = 2, c = –π, d = 0; b. x-axis ref: no; period: ,
x-inc: , p.s.: left, v.t.: none; c. LA: , RA: x = 0;
d. lkp: , mkp: , rkp: ; e.
5.Guide function:
a. a = 3, b = 2, c = , d = –1; b. amp of sine: 3, x-axis ref: no; period: π,
x-inc: , p.s.: right, v.t.: 1 down
c. ; d.
6.Guide function:
a. a = –2, b = 1, c = –π, d = 4; b. amp of cosine: 2, x-axis ref: yes;
period: 2π, x-inc: , p.s.: –π = left, v.t.: 4 up
c. (0, 6) ; d.
7.a.b.2.30 unitsc.
8.a.4 inchesb.after secc.frequency = 4 cycles per sec; period = sec
9.
10.; ; ; ;
11.
12.
13.verification of identity - answers may vary - see instructor if you would like your verification checked
14.verification of identity - answers may vary - see instructor if you would like your verification checked
15.verification of identity - answers may vary - see instructor if you would like your verification checked
16.verification of identity - answers may vary - see instructor if you would like your verification checked
17.verification of identity - answers may vary - see instructor if you would like your verification checked
18.verification of identity - answers may vary - see instructor if you would like your verification checked
19.
20.a. sin 40° b. tan 3x
21.
22.a. b. c.
23.a. sin 6y b. c.
24.; ;
25.; ;
26.; ;
27.35.a. b.
28.36.x = 90°, 210°, 330°
29.37.x = , , 2.0, 5.2
30.38.x = 45°, 135°, 225°, 315°
31.–45°39.
32.40.x = ,
33.41.x = , π,
34.42.x = , , ,
1