MAT146 Review for Test 4
1) Evaluate without a calculator. Find the exact value.
 1
a) sin 1   
 2

3

b) arccos 

2


 5 
e) sin 1  sin

4 

f) sin[tan-1(−2)]

3

c) tan 1  

3


12 

g) csc sin 1 
13 



d) tan 1  tan 
4

4

h) sec sin 1 
5

2) Verify that the equation is an identity.
a) sec2 θ(1 – sin2θ) = 1
b) (sin t + cos t)2 + 2sin(−t)cos(−t) = 1
cos 
cos 

 2 sec 
d)
1  sin  1  sin 
sin t
 csc t
e) cot t 
1  cos t
c) tan2θ – sin2θ =
sin 4 
cos 2 
1  (sin x  cos x) 2
f)
 cos x
2 sin x
g) sin(2x) = (tan x)(1 + cos(2x))
3) Find the exact value of each expression by using the appropriate addition or substitution identity.
 5 
a) cos 
 12 
 19 
b) sin 
 c) tan 255˚
 12 
 25 
d) cos
 e) tan 150˚ f) cos 165˚ g) sin 315˚
 12 
4) Find sin(x – y) and tan(x – y) exactly without a calculator.
2
1
sin x = , cos y =  , x is a quadrant II angle, y is a quadrant III angle
3
4
5) Find the exact value of sin 2x, cos 2x and tan 2x, if
2
tan x < 0
3
x
 x
x
6) Compute the exact values of sin   , cos  and tan   if
2
2
2
1
3
3

 x
  x 
a) cos x = 
b) tan x =
2
4
2
4
7) Find exact solutions over the indicated intervals.
a) sin x =
3
π/2 < x < π
5
a) 2sin x – 1 = 0 0  x  2
b) cos x =
b) 2sinθ +
3 = 0 all θ
8) Find all solutions: a) 2 sin2 θ + 5cos θ + 1 = 0 all θ
b) 4sin2 x – 3 = 0 0  x  2
c) 2cos2 x = −cos x on the interval [0, 2π]
9) Write the following expressions as a trigonometric function of one angle
a) cos 74˚ cos 44˚ + sin 74˚sin 44˚
b) sin 22˚cos 38˚ + cos 22˚sin 38˚
10) Number 75 in book
11) Solve each triangle
a) α = 67˚, β = 38˚, c = 49 meters
b) α = 15˚, b = 9.1 feet, c = 12 feet
c) γ = 121˚, c = 11cm, b = 4.2 cm d) a = 21.3 meters, b = 37.4 meters, c = 48.2 meters