# PROBLEMS INVOLVING TRIGONOMETRIC ANGLES FOR GRADE 11

Problems Involving Trigonometric Angles for Grade 11

Here we are going to see some example problems to show how to solve problems involving trigonometric angles.

sin C + sin D  =  2 sin [(C + D)/2] cos [(C - D)/2]

sin C - sin D  =  2 cos [(C + D)/2] sin [(C - D)/2]

cos C + cos D  =  2 cos [(C + D)/2] cos [(C - D)/2]

cos C - cos D  =  -2 sin [(C + D)/2] sin [(C - D)/2]

## Problems Involving Trigonometric Angles for Grade 11 - Examples

Question 1 :

Show that

(cosθ−cos3θ) (sin8θ+sin2θ)/(sin5θ−sinθ) (cos4θ−cos6θ)=1

Solution :

L.H.S

=  (cosθ−cos3θ) (sin8θ+sin2θ)/(sin5θ−sinθ) (cos4θ−cos6θ)

cosθ − cos3θ  =  2 sin 2θ sin θ  -------(1)

sin8θ + sin2θ  =  2 sin 5θ cos 3θ -------(2)

sin5θ − sinθ  =  2 cos 3θ sin 2θ -------(3)

cos4θ − cos6θ  =  2 sin 5θ sin θ -------(4)

(1)  (2)

=  4 sin 2θ sin θ sin 5θ cos 3θ  ------(A)

(3)  (4)

=  4 cos 3θ sin 2θ sin 5θ sin θ ------(B)

(A)/(B)

=  4 sin2θsinθsin 5θ cos 3θ / 4 cos 3θ sin 2θ sin 5θ sin θ

=  1

Hence proved.

Question 2 :

Prove that sin x + sin2x + sin3x = sin2x (1 + 2cos x).

Solution :

L.H.S

=  sin x + sin2x + sin3x

=  sin 2x + sinx + sin3x

=  sin 2x + 2 sin 2x cos x

=  sin 2x (1 + 2 cos x)

Question 3 :

Prove that

(sin 4x + sin2x) /(cos 4x + cos2x) = tan3x.

Solution :

sin 4x + sin2x  =  2 sin 3x cos x  --------(1)

cos 4x + cos2x  =  2 cos 3x cos x   --------(2)

(1) / (2)

=  2 sin 3x cos x / 2 cos 3x cos x

=  tan 3x

Question 4 :

Prove that

1 + cos2x + cos4x + cos6x = 4cos x cos 2x cos 3x.

Solution :

L.H.S:

=  1 + cos2x + cos4x + cos6x

=  1 + cos2x + 2 cos 5x cos x

=  2 cos2 x + 2 cos 5x cos x

=  2 cos x (cos x + cos 5x)

=  2 cos x (2 cos 3x cos 2x)

=  4 cos x cos 2x cos 3x

R.H.S

Hence proved.

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