Questions 1:
Find the exact value of sin (x / 2) if sin (x) = 1 / 4 and x is such that Pi / 2< x < Pi.
Solution to Question 1:

To find sin (x/2), we use the following half angle formula
sin (x/2) = + or  SQRT [ (1  cos x) / 2 ]

Since Pi < x < Pi / 2 then Pi / 2 < x / 2< Pi / 4 so that x/2 is in quadrant 1 and sin (x/2) is positive. Hence
sin (x/2) = SQRT [ (1  cos x) / 2 ]

Given that sin (x) = 1 / 4, we use the trigonometric identity sin ^{2}x + cos ^{2}x = 1 to find cos x, noting that x is in quadrant 2 and cos x is negative.
cos x =  sqrt(1  sin ^{2}x)
=  sqrt(1  1/16) =  sqrt(15) / 4

We now substitute cos x by its value in the formula for sin (x/2).
sin (x/2) = SQRT [ (1  sqrt(15) / 4) / 2 ]

Which may be simplified to
= (1/4) SQRT [ 8  2 SQRT(15) ]
Questions 2:
x is in quadrant 3, approximate sin (2 x) if cos (x) =  0.2. Round your answer to two decimal places.
Solution to Question 2:

sin (2x) may be calculated using the double angle trigonometric identity
sin (2x) = 2 sin (x) cos (x)

cos x is given, we need to find sin x using the identity sin ^{2}x + cos ^{2}x = 1 and noting that x is in quadrant 3 where sin x is negative
sin x =  SQRT[ 1  ( 0.2)^{2} ]

sin (2x) is now given by
sin (2x) = 2 [  SQRT[ 1  ( 0.2)^{2} ] ] (0.2)
= 0.39
Questions 3:
tan(x) = 4 and x is in quadrant III. Find the exact value of cos (x).
Solution to Question 3:

We first use the pythagorean identity 1 + tan ^{2}x = sec ^{2}x to find sec x in terms of tan x
sec (x) = + or  SQRT[ 1 + tan ^{2}x ]

Since x is in quadrant 3, sec (x) is negative. Hence
sec (x) =  SQRT[ 1 + 4 ^{2} ]
=  SQRT (17)

We now calculate cos (x) as follows
cos (x) = 1 / sec (x)
=  1 / SQRT (17)
Questions 4:
cos (2x) = 0.6 and 2x is in quadrant I. Find the exact value of csc (x).
Solution to Question 4:

We first use the identity cos(2x) = 2cos ^{2}x  1 to find cos (x)
0.6 = 2cos ^{2}x  1

Which gives
0.6 = 2cos ^{2}x  1
cos ^{2}x = 0.8

We now use the identity sin ^{2}x + cos ^{2}x = 1 to find sin x
sin x = SQRT(1  0.8)
csc (x) = 1 / SQRT(1  0.8)
Questions 5:
Find the exact value of cos (15^{o}).
Solution to Question 5:

Use the half angle formula cos (x/2) = + or  SQRT [ (1 + cos x) / 2 ] to write
cos (15^{o}) = SQRT [ (1 + cos 30^{o}) / 2 ]
= SQRT[ (1 + SQRT(3) / 2) / 2 ]
= (1/2) SQRT [ 2 + SQRT(3)]
Questions 6:
Find the exact value of tan ( 22.5^{o}).
Solution to Question 6:

Use the identity for negative to write
tan ( 22.5^{o}) =  tan (22.5^{o})

Use the half angle identity to tan (x/2) = sin x / (1 + cos x) to find tan(22.5) noting that 45/2 = 22.5
tan (22.5) = sin 45 / (1 + cos 45)
= ( SQRT(2) / 2 ) / [ 1 + SQRT(2)/2 ]
= SQRT(2)  1
Questions 7:
x and y are angles in quadrant 1 and 3 respectively and cos (x) = a and sin (y) = b. Find cos (x + y) in terms of a and b.
Solution to Question 7:

Use the sum formula cos (x + y) = cos x cos y  sin x sin y. We first need to find sin (x) and cos (y). x is in quadrant 1 where the sine is positive, Hence
sin (x) = SQRT (1  a^{2})

y is in quadrant 3 where the cosine is negative, hence
cos (y) =  SQRT(1  b^{2})

Finally
cos (x + y) = cos x cos y  sin x sin y
=  a SQRT(1  b^{2})  SQRT (1  a^{2}) b
Questions 8:
x is an angle in quadrant 3 and sin (x) = 1 / 3. Find sin (3x) and cos (3x).
Solution to Question 8:

Use the identity
sin(3x) = 3 sin x  4 sin^{3}x
= 3 (1/3)  4 (1 / 3)^{3}
= 23 / 27
Questions 9:
Reduce the power of the following trigonometric expression.
4 sin ^{3}(x) + 4 cos ^{3}(x)
Solution to Question 9:

Use the power reducing formulas sin ^{3}x = (3/4) sinx  (1/4) sin (3x) and cos ^{3}x = (3/4) cos x+ (1/4) cos(3x) to write that
4 sin ^{3}(x) + 4 cos ^{3}(x)
= 4 [ (3/4) sinx  (1/4) sin (3x) ] +
4 [ (3/4) cos x + (1/4) cos(3x) ]
= 3 sinx  sin (3x) + 3 cos x + cos 3x
Questions 10:
Factor the following trigonometric expression.
sin (x) + sin (2x)
Solution to Question 10:

Use the identity sin (2x) = 2 sin x cos x to write
sin (x) + sin (2x) = sin x + 2 sin x cos x
= sin x (1 + 2 cos x)
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