THE COMPOUND ANGLE IDENTITIES

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1 TRIGONOMETRY THE COMPOUND ANGLE IDENTITIES

2 Question 1 Prove the validity of each of the following trigonometric identities. a) sin x + cos x 4 4 b) cos x sin x + 2cos x 3 3 c) cos 2x + + cos 2x cos 2x 3 3 d) sin ( x + y) tan x + tan y cos x cos y e) tan x + tan x 1 4 4

3 Question 2 Prove the validity of each of the following trigonometric identities. a) sin x + 3 cos x + 2sin x 3 3 b) cos x sin x cos( x + y) sin y cos y sin y cos y tan x tan x c) tan ( x 60 ) tan ( x 60 ) d) sin ( + ) sin( ) cos cos x y x y y x e) cot ( x y) cot x cot y 1 + cot x + cot y

4 Question 3 Prove the validity of each of the following trigonometric identities. 2 2 a) cos( + ) cos( ) cos sin x y x y x y P + Q P Q b) sin P sin Q 2 cos sin 2 2 c) 2 2 sin θ + + sin θ d) cos x cos x + sin x tan 2x cos 2x P + Q P Q e) cos P + cosq 2 cos cos 2 2

5 Question 4 If sin ( ) θ + α = 2sinθ, show clearly that sinα tanθ =. 2 cosα Question 5 By expanding tan ( θ + 45 ) with a suitable value for θ, show clearly that tan 75 = Question 6 By expanding sin ( 45 x) with a suitable value for x, show clearly that cosec15 =

6 Question 7 By expanding tan ( θ + 45 ) with a suitable value for θ, show clearly that tan105 = 2 3. Question 8 By expanding cos( y + 45 ) with a suitable value for y, show clearly that sec 75 = Question 9 By considering the expansion of tan ( A B) clearly that + with suitable values for A and B, show cot 75 = 2 3.

7 Question 10 Show clearly, by using the compound angle identities, that 6 2 sin15 =. 4 Question 11 Show clearly, by using the compound angle identities, that 2 6 cos105 =. 4 Question 12 Show clearly, by using the compound angle identities, that tan15 = 2 3.

8 Question 13 ( ) sin A + B sin Acos B + cos Asin B. a) Use the above trigonometric identity with suitable values for A and B, to show that sin 75 =. 4 b) Hence by using the trigonometric expansion of cos( 75 α ) value for α, show clearly that cos165 = sin with a suitable Question sin A = and 13 4 cos B =. 5 If A is obtuse and B is acute, show clearly that sin 33 + =. 65 ( A B)

9 Question 15 8 sinθ = and 17 5 cosϕ =. 13 If θ is obtuse and ϕ is acute, show clearly that cos =. 221 ( θ ϕ ) Question 16 The constants a and b are such so that 1 tan a = and 3 1 tan b =. 7 Determine the exact value of cot ( a b), showing all the steps in the workings. cot ( a b) = 11 2

10 Question sin x = and cos y =. 17 If x is obtuse and y is acute, show clearly that sin 220 =. 221 ( x y) Question 18 8 sin P = and 17 4 tan Q =. 3 If P is obtuse and Q is reflex, show clearly that cos 13 =. 85 ( P Q)

11 Question 19 5 sinθ = and 13 7 sinϕ =. 25 If θ is obtuse and ϕ is such so that 180 < ϕ < 270, show clearly that sin 36 + =. 325 ( θ ϕ ) Question 20 3 cosθ = and 5 24 tanϕ =. 7 If θ is reflex, and ϕ is also reflex, show clearly that sin 44 =. 125 ( θ ϕ )

12 Question sinθ = and cosϕ =. 17 If θ is obtuse and ϕ is reflex, show clearly that sec =. 87 ( θ ϕ )

13 Question 22 3 cot A = and 4 5 cos B =. 13 If A is reflex and B is also reflex, show clearly that tan 56 + =. 33 ( A B)

14 Question 23 1 sin A = and 3 1 cos B =. 2 If A is obtuse and B is reflex, show clearly that sin =. 6 ( A B) Question 24 The point A lies on the y axis above the origin O and the point B lies on the y axis below the origin O. The point C ( 12,0) is at a distance of 20 units from A and at a distance of 13 units from B. By considering the tangent ratios of angle ACB is exactly OCA and OCB, show that the tangent of the

15 Question 25 Solve each of the following trigonometric equations. a) ( θ ) cos + 30 = sinθ, 0 θ < 360 b) 3cos( x 30 ) sin ( x 60 ) + =, 0 x < 360 c) sin ( y 30 ) sin ( y 45 ) = +, 0 y < 360 d) sin ( ϕ 30 ) cos( ϕ 45 ) + =, 0 ϕ < 360 e) cos ( α 60 ) cos( α 45 ) =, 0 α < 360 θ = 30, 210, x = 60, 240, y = 82.5, 262.5, ϕ = 52.5, 232.5, α = 52.5, 232.5

16 Question 26 Solve each of the following trigonometric equations. a) ( θ ) sin 45 = sinθ, 0 θ < 360 b) cos ( x 30 ) sin ( x 30 ) = +, 0 x < 360 c) cos ( y 30 ) sin ( y 45 ) = +, 0 y < 360 d) sin ( ϕ 30 ) cos ( ϕ 45 ) =, 0 ϕ < 360 e) cos ( α 60 ) cos( α 60 ) = +, 0 α < 360 θ = 112.5, 292.5, x = 45, 225, y = 37.5, 217.5, ϕ = 82.5, 262.5, α = 0, 180

17 Question 27 Solve each of the following trigonometric equations. a) sin θ + = sinθ, 0 θ < 2 4 b) c) 2 cos x + = cos x + 6 3, 0 θ < 2 5 sin y = cos y + 3 6, 0 y < 2 (very hard) d) 2cos ϕ + + sin ϕ + = 0, 0 ϕ < e) 2 cos α + = sin α + 4 6, 0 α < θ =,, x =,, y =,, ϕ =,, α =,

18 Question 28 Solve each of the following trigonometric equations. a) sin ( θ 20 ) sin ( θ 60 ) = +, 0 θ < 360 b) cos( x 35 ) cos( x 55 ) =, 0 x < 360 c) sin ( y 48 ) cos( y 12 ) = +, 0 y < 360 d) sin ( ϕ 72 ) cos( ϕ 38 ) + =, 0 ϕ < 360 e) cos( α 36 ) cos( α 72 ) =, 0 α < 360 θ = 70, 250, x = 45, 225, y = 63, 243, ϕ = 28, 208, α = 54, 234

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