Question 1

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 ²x + cos ²x = 1 to find cos x, noting that x is in quadrant 2 and cos x is negative.
  
  cos x = - sqrt(1 - sin ²x)
  
  = - 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) ]
  
  

Question 2

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 ²x + cos ²x = 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

Question 3

Solution to Question 3:

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

Question 4

Solution to Question 4:

• We first use the identity cos(2x) = 2cos ²x - 1 to find cos (x)
  
  0.6 = 2cos ²x - 1
  
  
• Which gives
  
  0.6 = 2cos ²x - 1
  
  cos ²x = 0.8
  
  
• We now use the identity sin ²x + cos ²x = 1 to find sin x
  
  sin x = SQRT(1 - 0.8)
  
  csc (x) = 1 / SQRT(1 - 0.8)

Question 5

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)]

Question 6

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
  
  

Question 7

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 - a²)
  
  
• y is in quadrant 3 where the cosine is negative, hence
  
  cos (y) = - SQRT(1 - b²)
• Finally
  
  cos (x + y) = cos x cos y - sin x sin y
  
  = - a SQRT(1 - b²) - SQRT (1 - a²) b

Question 8

Solution to Question 8:

• Use the identity
  
  sin(3x) = 3 sin x - 4 sin³x
  
  = 3 (1/3) - 4 (1 / 3)³
  
  = 23 / 27

Question 9

Solution to Question 9:

• Use the power reducing formulas sin ³x = (3/4) sinx - (1/4) sin (3x) and cos ³x = (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

Question 10

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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