__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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