PROBLEMS INVOLVING TRIGONOMETRIC ANGLES FOR GRADE 11

Problem 1 :

Show that

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

Solution :

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.

Problem 2 :

Prove that :

sin x + sin2x + sin3x  =  sin2x (1 + 2cos x)

Solution :

sin x + sin2x + sin3x :

=  sin 2x + 2 sin 2x cos x

=  sin 2x (1 + 2 cos x)

Problem 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

Problem 4 :

Prove that

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

Solution :

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

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