PROVE THAT PROBLEMS USING COMPOUND ANGLES FORMULA

Problem 1 :

Prove that (i) cos(30° + x)  =  (√3 cos x − sin x)/2

(ii) cos(π + θ) = −cos θ

(iii) sin(π + θ) = −sin θ.

Solution :

(i) cos(30° + x) :

cos (A + B)  =  cos A cos B - sin A sin B

cos(30° + x)  =  cos 30 cos x - sin 30 sin x

  =  (√3/2) cos x - (1/2) sin x

  =  (√3 cos x - sin x)/2

(ii) cos(π + θ) :

(π + θ) lies in the 3rd quadrant 

In third quadrant, we will have negative sign for all trigonometric ratios other than tan θ and cot θ. Here we have cos θ.

So, cos(π + θ)  =  -cos θ

(iii) sin(π + θ) = −sin θ :

(π + θ) lies in the 3rd quadrant 

In third quadrant, we will have negative sign for all trigonometric ratios other than tan θ and cot θ. Here we have sin θ.

So, sin(π + θ)  =  -sin θ

Problem 2 :

Find a quadratic equation whose roots are sin 15° and cos 15°.

Solution :

Since the roots of the required quadratic equation are sin 15° and cos 15°.

let α  =  sin 15° and β  = cos 15°

α  =  sin 15°  =  sin (45 - 30)

sin (A - B)  =  sin A cos B - cos A sin B

  =  sin 45 cos 30 - cos 45 sin 30

  =  (1/√2) (√3/2) - (1/√2) (1/2)

  =  (√3/2√2) - (1/2√2)

α = sin 15°  =  (√3 - 1)/2√2  ---(1)

α  =  cos 15°  =  cos (45 - 30)

cos (A - B)  =  cos A cos B + sin A sin B

  =  cos 45 cos 30 + sin 45 sin 30

  =  (1/√2) (√3/2) + (1/√2) (1/2)

  =  (√3/2√2) + (1/2√2)

β = cos 15°  =  (√3 + 1)/2√2  ---(2)

α + β

αβ

General form of quadratic equation

x2 + (α + β)x + α β  =  0

x2 + (√3/√2)x + (1/4)  =  0

4x2+ 2√3x + 1  =  0 

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