Trigonometric Functions
Questions With Answers

A set of trigonometry questions related to trigonometric functions are presented. The solutions and answers are provided.



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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 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) ]

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 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

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 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)

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 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)

Questions 5:

Find the exact value of cos (15o).

Solution to Question 5:

  • Use the half angle formula cos (x/2) = + or - SQRT [ (1 + cos x) / 2 ] to write

    cos (15o) = SQRT [ (1 + cos 30o) / 2 ]

    = SQRT[ (1 + SQRT(3) / 2) / 2 ]

    = (1/2) SQRT [ 2 + SQRT(3)]

Questions 6:

Find the exact value of tan (- 22.5o).

Solution to Question 6:

  • Use the identity for negative to write

    tan (- 22.5o) = - tan (22.5o)

  • 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 - 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

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 sin3x

    = 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 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

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)

More math problems with detailed solutions in this site.


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Updated: 2 April 2013

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