Download Trig Exam 2 Review

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 of every 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 the departmental 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 y = −3 sin (2x - π ) + 1
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 y =
a.
Find: a = __________ b = __________ c = __________ d = __________
b.
Find: amp: _______________ x-axis reflection: ______________ period: ______________
1
π⎞
⎛
cos ⎜ x + ⎟ − 2
4
3⎠
⎝
x-increment: ______________ phase shift: _______________ vertical translation: _______________
c.
Find: 5 key points:
____________ ____________ ____________ ____________ ____________
d.
Manually graph two periods of the given function.
3.
π⎞
⎛
Given y = 2 tan ⎜ x − ⎟ + 3
4⎠
⎝
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.
1
Trig Exam 2 Review F07 O’Brien
1
cot (2x + π )
2
4.
Given y =
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.
π⎞
⎛
Given y = 3 csc ⎜ 2x - 4 ⎟ − 1
⎝
⎠
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 y = −2 sec (x + π ) + 4
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 =
5
and tan s < 0, find sin s.
5
10.
Given sin θ = −
4
, cos θ < 0 , find the remaining five trigonometric functions of θ.
5
11.
Use identities to write sec x in terms of sin x.
2
Trig Exam 2 Review F07 O’Brien
1 + tan 2 x
12.
Write
13.
Verify the identity
14.
Verify the identity 2cos A − sec A = cos A −
15.
Verify the identity tan θ + cot θ = sec θ csc θ.
16.
Verify the identity
17.
Verify the identity (sec x − tan x )2 =
18.
Verify the identity cos 2 A + tan 2 A − 1 = tan 2 A ⋅ sin 2 A .
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.
1 + cot 2 x
in terms of sine and cosine, and simplify so that no quotients appear in the final expression.
1
3
cos x + 3 sin x
+
=
.
sin x cos x
sin x cos x
tan A
.
csc A
sin x
= csc x + cot x .
1 − cos x
a. sin 25° cos 15° + cos 25° sin 15°
1 − sin x
.
1 + sin x
b.
tan 2x + tan x
1 − tan 2x ⋅ tan x
21.
⎛θ
⎞
Find an angle that makes sec θ = csc ⎜ + 20 o ⎟ .
⎝2
⎠
22.
Given cos α =
a. sin (α − β )
8
24
, α in Quadrant IV, and sin β = −
, β in Quadrant III, find
17
25
b. cos(α − β )
c. tan (α − β )
23.
Write each expression in terms of a single trigonometric function:
2 tan 4 θ
2
2
c.
a. 2 sin 3y cos 3y
b. cos 6α −sin 6α
1 − tan 2 4 θ
24.
Use the half-angle identities to find the exact values of the sine, cosine, and tangent of
25.
Given cos θ =
40
with θ in Quadrant IV, find the sine, cosine, and tangent of 2θ .
41
26.
Given cot α =
8
α
with α in Quadrant III, find the sine, cosine, and tangent of .
15
2
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 sin
7π
.
12
3θ
θ
+ sin as the product of two functions.
4
2
3
Trig Exam 2 Review F07 O’Brien
7π
π
− cos
. Do not use a calculator.
12
12
30.
Evaluate cos
31.
⎛
2 ⎞⎟
. Do not use a calculator.
Find the exact degree value of θ = sin −1 ⎜ −
⎜ 2 ⎟
⎝
⎠
32.
Use a calculator to find the degree measure of θ = arccot (–.3451).
33.
⎛ 1⎞
Find the exact radian value of y = cos −1 ⎜ − ⎟ . Do not use a calculator.
⎝ 2⎠
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.
⎛
⎛ 5 ⎞⎞
⎛
−1 1
−1 3 ⎞
b. tan ⎜ cos 2 − sin 4 ⎟
a. cos ⎜⎜ arcsin ⎜ − 13 ⎟ ⎟⎟
⎝
⎠⎠
⎝
⎠
⎝
36.
Find the exact solutions of 2 sin x – 1 = csc x in the interval [0°, 360°).
37.
Find the exact solutions of tan 2 x + tan x = 2 in the interval [0, 2π).
38.
2
2
Find the exact solutions of cos x − sin x = 0 in the interval [0°, 360°).
39.
Find the exact solutions of tan x +
3 = sec x in the interval [0, 2π).
40.
Find the exact solutions of 2 3 sin
θ
= 3 in the interval [0, 2π).
2
41.
Find the exact solutions of cos 2x + cos x = 0 in the interval [0, 2π).
42.
Find the exact solutions of sin x ⋅ cos x =
1
in the interval [0, 2π).
4
Answers
1.
2.
a. a = –3, b = 2, c = π, d = 1; b. amp: 3, x-axis ref: yes; period: π,
π
2π
, p.s.:
right, v.t.: 1 up
x-inc:
4
4
⎛ 2π ⎞ ⎛ 3π
⎞ ⎛ 4π ⎞ ⎛ 5π ⎞ ⎛ 6π ⎞
c. ⎜ 4 , 1⎟ ⎜ , − 2 ⎟ ⎜ , 1⎟ ⎜ , 4 ⎟ ⎜ , 1⎟ ; d.
⎝
⎠ ⎝ 4
⎠ ⎝ 4 ⎠ ⎝ 4
⎠ ⎝ 4
⎠
π
1
1
, b = 1, c = − , d = –2; b. amp: , x-axis ref: no;
3
4
4
π 3π
π
2π
=
period: 2π, x-inc:
, p.s.: − = −
left, v.t.: 2 down
2
6
3
6
⎞ ⎛ 4π
⎞ ⎛ 10π
7⎞ ⎛π
9 ⎞ ⎛ 7π
7⎞
⎛ 2π
, − ⎟ ⎜ 6 , − 2⎟ ⎜ , − ⎟ ⎜ , − 2⎟ ⎜
, − ⎟ ; d.
c. ⎜ −
6
⎝
⎠
⎝
⎠
6
4
6
4
6
4⎠
⎝
⎠
⎝
⎠
⎝
a. a =
4
Trig Exam 2 Review F07 O’Brien
3.
4.
5.
6.
π
, d = 3; b. x-axis ref: no; period: π,
4
π
π
π
3π
, p.s.:
right, v.t.: 3 up; c. LA: x = − , RA: x =
;
x-inc:
4
4
4
4
⎛π ⎞
⎛ 2π ⎞
d. lkp: (0, 1), mkp: ⎜ 4 , 3 ⎟ , rkp: ⎜ 4 , 5 ⎟ ; e.
⎝
⎠
⎝
⎠
a. a = 2, b = 1, c =
π
1
, b = 2, c = –π, d = 0; b. x-axis ref: no; period:
,
2
2
π
4π
4π
π
left, v.t.: none; c. LA: x = −
, RA: x = 0;
, p.s.: − = −
x-inc:
2
8
8
8
⎛ 3π 1 ⎞
⎛ 2π ⎞
⎛ π 1⎞
d. lkp: ⎜ − 8 , 2 ⎟ , mkp: ⎜ − 8 , 0 ⎟ , rkp: ⎜ − 8 , − 2 ⎟ ; e.
⎝
⎠
⎝
⎠
⎝
⎠
a. a =
π⎞
⎛
Guide function: y = 3sin ⎜ 2x − 4 ⎟ − 1
⎝
⎠
π
a. a = 3, b = 2, c = , d = –1; b. amp of sine: 3, x-axis ref: no; period: π,
4
π
π 2π
right, v.t.: 1 down
=
, p.s.:
x-inc:
8
4
8
⎞ ⎛ 5π
⎞ ⎛ 7π
⎞ ⎛ 9π
⎞
⎛π
⎞ ⎛ 3π
c. ⎜ 8 , - 1⎟ ⎜ 8 , 2 ⎟ ⎜ 8 , - 1⎟ ⎜ 8 , - 4 ⎟ ⎜ 8 , - 1⎟ ; d.
⎝
⎠ ⎝
⎠ ⎝
⎠ ⎝
⎠ ⎝
⎠
Guide function: y = −2cos(x + π ) + 4
a. a = –2, b = 1, c = –π, d = 4; b. amp of cosine: 2, x-axis ref: yes;
π
2π
period: 2π, x-inc:
, p.s.: –π = −
left, v.t.: 4 up
2
2
⎛ π ⎞ ⎛ 2π
⎛ 2π
⎞ ⎛ π ⎞
⎞
c. ⎜ −
, 2 ⎟ ⎜ − , 4 ⎟ (0, 6) ⎜ , 4 ⎟ ⎜ , 2 ⎟ ; d.
⎝ 2
⎠ ⎝ 2 ⎠
⎠
⎝2 ⎠ ⎝ 2
7.
a.
s(t ) = − 6 cos
8.
a.
4 inches
9.
sin s = −
π
t
2
b.
2.30 units
b.
after
1
sec
8
c.
1
4
c. frequency = 4 cycles per sec; period =
1
sec
4
2 5
5
5
Trig Exam 2 Review F07 O’Brien
10.
3
4
5
3
5
cos θ = − ; tan θ = ; cot θ = ; sec θ = − ; csc θ = −
5
3
3
4
4
11.
sec x =
12.
13.
14.
15.
16.
17.
18.
tan 2 x
verification of identity - answers may vary - see instructor if you would like your verification checked
verification of identity - answers may vary - see instructor if you would like your verification checked
verification of identity - answers may vary - see instructor if you would like your verification checked
verification of identity - answers may vary - see instructor if you would like your verification checked
verification of identity - answers may vary - see instructor if you would like your verification checked
verification of identity - answers may vary - see instructor if you would like your verification checked
− 6− 2
cos 195° =
4
a. sin 40° b. tan 3x
140 o
3
297
304
297
b. cos(α − β ) =
c. tan (α − β ) =
a. sin (α − β ) =
425
425
304
a. sin 6y b. cos 12 α c. tan 8θ
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
± 1 − sin 2 x
1 − sin 2 x
2+ 3
7π
7π − 2 − 3
=−2− 3
=
; cos
; tan
12
2
12
2
720
1519
720
sin 2θ = −
; cos 2θ =
; tan 2θ = −
1681
1681
1519
sin
7π
=
12
sin
α 5 34
α − 3 34
α −5
=
; cos =
; tan =
2
34
2
34
2
3
1
[cos 10x + cos 4x ]
2
− 2
4
5θ
θ
2sin
cos
8
8
36.
x = 90°, 210°, 330°
37.
x=
x = 45°, 135°, 225°, 315°
−
6
2
38.
31.
–45°
39.
32.
θ ≈ 109.4990544 o
40.
33.
y=
34.
5 6
12
41.
42.
b.
3 7 −4 3
5
a.
30.
2π
3
12
13
35.
π 5π
,
, 2.0, 5.2
4 4
11π
6
2π 4π
,
x=
3
3
π
π
x = , π,
3
3
x=
x=
π 5π 13π 17π
,
,
,
12 12 12
12
6