GRADE 11 TRIGONOMETRY PROBLEMS INVOLVING ANGLES

Problem 1 :

prove that

sin θ/2 sin 7θ/2 + sin 3θ/2 sin 11θ/2 = sin2θ sin 5θ.

Solution :

L.H.S :

  =  sin θ/2 sin 7θ/2 + sin 3θ/2 sin 11θ/2

  =  (1/2) [2 sin θ/2 sin 7θ/2] + (1/2) [2 sin3θ/2 sin 11θ/2]

2 sin θ/2 sin 7θ/2  =  cos 3θ - cos 4θ   

2 sin3θ/2 sin 11θ/2  =  cos 4θ - cos 7θ

 Applying the those values, we get

  =  (1/2) [cos 3θ - cos 4θ] + (1/2) [cos 4θ - cos 7θ]

  =  (1/2) [cos 3θ - cos 4θ + cos 4θ - cos 7θ]

  =  (1/2) [cos 3θ - cos 7θ]

  =  (1/2) [2 sin 5θ sin 2θ]

  =  sin 5θ sin 2θ

Hence proved.

Problem 2 :

Prove that

cos (30°−A) cos (30°+A) + cos (45°−A) cos (45°+A) = cos2A +  (1/4)

Solution :

L.H.S

  =  cos (30°−A) cos (30°+A) + cos (45°−A) cos (45°+A)

 =  (1/2) (2cos (30°−A) cos (30°+A)) + (1/2) (2cos (45°−A) cos (45°+A))

  =  (1/2) [cos 60 + cos 2A + cos 90 + cos 2A]

  =  (1/2) [(1/2) + cos 2A + 0 + cos 2A]

  =  (1/2) [(1/2) + 2cos 2A]

  =  1/4 + cos 2 A

Hence proved.

Problem 3 :

Prove that

(sin x + sin3x + sin5x + sin7x) / (cos x + cos3x + cos5x + cos7x)  =  tan4x.

Solution :

  =  (sin x + sin3x + sin5x + sin7x) / (cos x + cos3x + cos5x + cos7x) 

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

sin5x + sin7x  =  2 sin 6x cos x-------(2)

cos x + cos3x  =  2 cos 2x cos x -------(3)

cos5x + cos7x  =  2 cos 6x cos x -------(4)

(1) + (2)

  =  2 sin 2x cos x + 2 sin 6x cos x

  =  2 cos x (sin 2x + sin 6x)  -----(A)

(3) + (4)  

  =   2 cos 2x cos x + 2 cos 6x cos x

  =  2 cos x (cos 2x + cos 6x)  -----(B)

(A)/(B)

  =  2 cos x (sin 2x + sin 6x) / 2 cos x (cos 2x + cos 6x)

  =  (sin 2x + sin 6x) / (cos 2x + cos 6x)

Again using the formula for sin C + sin D and cos C + cos D, we get

  =  2 sin 4x cos 2x / 2 cos 4x cos 2x

  =  tan 4x

Hence proved. 

After having gone through the stuff given above, we hope that the students would have understood how to solve trigonometry problems involving angles"

Apart from the stuff given in this section, if you need any other stuff in math, please use our google custom search here.

Kindly mail your feedback to v4formath@gmail.com

We always appreciate your feedback.