Math 170 – Trigonometry
Sample Big Test #3
ANSWERS
(The answers are not guaranteed to be correct)
1.
Evaluate each of these expressions exactly, without using a calculator (except for basic
arithmetic). A diagram (or two) may be helpful.
3
21
a)
sin2arctan7
29

 
b)
2
cos2arccos5

 
c)
3
-1
cosarcsin5 + arccos 2 

 
 
-17
25
-
4 +3 3
10
3 + 5sin = 3sin.
-60° + n360° ; 240° + n360° ; where n is an integer
2.
Find all degree solutions of this equation:
3.
Find all radian solutions such that 0  x < 2 :
π
3π
π
5π
x= 2 ;x= 2 ;x= 6 ;x= 6
4.
Find all solutions (exact values only) of this equation: 3sinx - 10cotx = -9cscx
cosx – 2sinxcosx = 0

61 - 5 
 + 2kπ where k is an integer
3


x = ±arccos
2sinθ = sin2θ
5.
Solve this equation for θ if 0 ≤ θ < 360:
θ = 0° θ = 180°
6.
Find all solutions in radians using exact values only.
-1
sin2xcos3x + cos2xsin3x = 2
π
π
x = 30 (7 + 12k) or x = 30 (11 + 12k) where k is an integer
7.
Find all solutions if 0 ≤ θ < 360. If necessary, round your answers to the nearest
tenth of a degree.
2
3cos 3θ + 10cos3θ – 12 = 0
θ = 23.2° θ = 143.2° θ = 263.2° θ = 96.8° θ = 216.8° θ = 336.8°
8.
The bell tower of the cathedral in Pisa, Italy leans 5.6 from the vertical. A tourist
stands 105 m from its base, with the tower leaning directly towards her. She measures
the angle of elevation to the top of the tower to be 29.2. Find the length of the tower.
Give both an exact answer (i.e. before you have used your calculator) and one rounded
to the nearest meter. (Please draw a figure and label it well.)
length of tower =
105 sin29.2°
meters (exactly)
sin66.4°
length of tower ≈ 56 meters
The length of the leaning tower of Pisa is about 56 meters.
9.
The following information refers to triangle ABC. Sketch the triangle and then solve it
using the Law of Sines.
∠A = 23 ,
10.
∠B = 110, c = 50
∠C = 47°
a ≈ 47 and b ≈ 64
Show that this is an identity by transforming one side into the other. Justify each of
your steps.
2
1 - tan 
cos2 =
2
1 + tan 
I decided to begin with the Right Hand Side, though this might not be the best way.
1 - tan2θ
1 + tan2θ
Right hand side
sin2θ
1cos2θ
sin2θ
1+
cos2θ
sinx
tanx = cosx
cos2θ - sin2θ
cos2θ + sin2θ
Multiply by 1 in the form of
cos2θ cos2x – sin2x = cos2x, and cos2x + sin2x = 1
cos2θ
cos2θ
11.
-3
If sin = 7
a)
sin2
b)

sin2
 
12.
Prove the following identity. Be sure to show your justifications for each step.
and
180 <  < 270 , find the following. (Show all of your work.):
-12 10
49
7 - 2 10
14
sin( + ) + sin( – ) = 2sin cos
[sinα cosβ + cosα sinβ ] + [sinα cosβ – cosα sinβ ]
sin(x + y) = sinxcosy + cosxsiny
sin(x – y) = sinxcosy – cosxsiny
2sinα cosβ
add similar terms
13.
The following information refers to triangle ABC. Sketch the triangle and then solve it
using the Law of Sines.
∠A = 62.1 , a = 7.31 feet,
b = 8.11 feet
This is the ambiguous case. You will not have the ambiguaous case on BT #3.
B1 ≈ 78.7° C1 ≈ 39.2° c1 ≈ 5.23 feet
B2 ≈ 101.3° C2 ≈ 16.6° c2 ≈ 2.36 feet
Problems 14 & 15 use the Law of Cosines which will not be on Big Test #3.
14.
The following information refers to triangle ABC. Sketch the triangle and then solve it
using the Law of Cosines. Give the angles to the nearest degree.
a = 66 cm, b = 120 cm, c = 72 cm
15.
∠A ≈ 28° ∠B ≈ 121°
∠C ≈ 31°
The following information refers to triangle ABC. Sketch the triangle and then solve it
using the Law of Cosines. Give the angles to the nearest degree and the sides to the
nearest inch.
a = 45 inches, ∠B = 42, c = 33 inches
∠A ≈ 91° ∠C ≈ 47° b ≈ 30 inches