# Trigonometric Functions Questions With Answers

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

 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.