Review Guide for test 2. The emphasis of this guide is on parts of section 6.5 through sec. 7.4. All material
covered previously is also fair game. As always, make sure you can do all of the homework and other
assigned problems and activities.
1. Find the amplitude, period, phase shift, and vertical shift of
, and graph. Make sure
the scale you use is clearly marked on each axis. Draw at least two cycles. (sec 6.6)
2. Find the period and sketch the graph of two consecutive cycles of
. Make sure the
scale you are using on both axis is clearly labeled. Include asymptotes in your graph. (sec 6.5 & 6.6)
3. Write a trig equation that would give the following graph. (sec 6.5 & 6.6)
4.) Suppose the water level in feet at the Boston Harbor can be described
by an equation of the form
where x = 0 corresponds
to midnight. In a 24 hour period there are two complete cycles and at 2
A.M. the water is at its highest depth of 12 feet, while at 8 A.M. the water
level is at its lowest depth of 3 feet. Find a, b, c, and d. (sec. 6.6)
. Show a dotted boundary
5.) Graph
graph and label at least 3 points exactly. Make sure you can label at least one point that is not on an axis. (sec.
6.6)
6.) Verify the following identity: sin 2θ(1 + cot 2θ) = 1. (sec. 7.3)
7.) Verify the identity: tan θ + cot θ = 2csc 2θ (sec. 7.3)
8.) Find the exact values of each of the following: (sec. 7.1 & 7.2)
a.) tan105 °
b.) sec(-5π /12)
c.) cos (3π /8)
d.) sin(5π /24)
e.) sin(2x) if cosx = -3/5 and sinx > 0
f.) sin(x/2) if cosx = -3/5 and sinx < 0 , 0° < x < 360°
g.) cos(θ +3π /2) if cos θ = 1/3 and sin θ < 0.
h.) sin(π /2 -θ ) if cos θ = 1/3 and sin θ < 0.
9.) Find the exact values of each of the following: (review from unit 1)
a.) cos(-945°)
b.) secq if tan θ = 16 and sin θ < 0
c.) cos(θ + 3π/2) if cos θ = 1/3 and sin θ < 0.
d.) sin(π/2 - θ) if cos θ = 1/3 and sin θ < 0.
10.) Let P(-5, -12) be on the terminal side of an angle in standard position with positive measure q , 0 < θ <
2π. Find each of the following: (sec. 7.2)
a.) cos θ
b.) sin 2θ
c.) tan(θ/2)
11.) Verify the following identity: cos x - (tan y sin x) = sec y cos (x + y) (sec. 7.3)
12.) Write sec t in terms of cot t if 0 < t < π /2 . (sec. 7.1)
13.) Find all values of x in the interval [-2π, 2π] such that
answers.
in the equation y = cos x. Give exact
14.) Find the exact values of the following: (sec. 7.4)
a. tan-1(√3/3)
b. cos-1(π/2)
c.
d.
e.
f.
g.
h.
15.) Write as an algebraic expression, assuming x > 0 : cos(2tan-1x) (sec. 7.4)
16.) Sketch the graph of the equation
y = sin-1(x – 2) + π/2 (sec. 7.4 and transformations from Math 121)
17.) Find all values of x in the interval [-2π, 2π] such that y > √3/2 in the equation y = cos x. Give exact
answers.
18.) Find all solutions in [-π, π] of sin t = 0.46.
Answers:
1.) amp = 1
2.) per = 6π
per = 8/3 phase shift = 1/3 vertical shift = -2
ph. sh. = -π
3.) one solution is:
4.) a = -9/2, b = π /6, c = -5π /6, d = 15/2
5.)
6.) sin2θ(1 + cot2θ) = 1
sin2θ + sin2θ cot2θ = 1
sin2θ + cos2θ = 1
7.)
8.) (a.)
(b.)
(d.)
(e.)
9.) a.) -√(2)/2
10.) (a.) -5/13
(c.)
(f.)
(g.)
h.) 1/3
c.) -2√(2)/3
b.)
(b.) 120/169
d.) 1/3
(c.) -3/2
11.)
12.)
13.)
14.) (a.) π/6, (b.) none, (c.) -1/3, (d.) π/6
15.)
(e.) √(65)/4, (f.) 0, (g.) 3/√(10), (h.) 120/169
16.)
17.) [-2π, -11π/6) ∪ (-π/6, π/6) ∪ (11π/6, 2π]
18.) 0.4780, 2.6636
[-2π, -11π/6)∪(-π/6,π/6)∪(11π/6,2π]