Question

if sin phi, sin theta, cos phi are in G.P, and cos 2theta = kcos^2 (pi/4 + phi) , the value of k/8​

Answer 1

Given :-

sin Ø , sin θ , cos Ø are the G.P.

cos² θ = k cos² [( π/4) + Ø].

To find :-

The value of k/8.

Solution :-

Given that

sin Ø , sin θ , cos Ø are the G.P.

We know that

If a, b, c are in the GP then b² = ac

So,

sin² θ = sin 2Ø × cos Ø

On multiplying with 2 both sides then

=> 2 sin² θ = 2 sin Ø × cos Ø

=> 2 sin² θ = sin 2Ø

Since, sin 2A = 2 sin A cos A

=> 2 sin² θ = sin 2Ø

=> 2( 1- cos² θ) = sin 2Ø

Since, sin² A + cos² A = 1

=> 2 - 2 cos² θ = sin 2Ø

=> 1 + 1 - 2 cos² θ = sin 2Ø

=> 1+ (1 - 2 cos² θ) = sin 2Ø

=> 1 -(2 cos² θ -1) = sin 2Ø

=> 1 - cos 2θ = sin 2Ø

Since, cos 2A = 2 cos² A -1

=> 1 - cos 2θ = - cos ( π/2 + 2Ø)

=> - cos 2θ = 1 - cos ( π/2 + 2Ø)

=> cos 2θ = -1 +cos ( π/2 + 2Ø)

=> cos 2θ = cos [( π/2) + 2Ø]+1

=> cos 2θ = cos 2[( π/4) + Ø]+1

=> cos 2θ =2 cos² [( π/4)+ Ø]-1+1

since, cos 2A = 2 cos² A -1

where , A = [(π/4)+Ø]

=> cos 2θ = 2 cos² [( π/4)+ Ø]

Given that

cos 2θ = k cos² [( π/4)+ Ø]

on comparing them

we have , k = 2

Now, The value of k/8 = 2/8 = 1/4

Answer :-

The value of k/8 = 1/4

Used formulae:-

→ If a, b, c are in the GP then b² = ac

→ sin² A + cos² A = 1

→ cos 2A = 2 cos² A -1

→ sin 2A = 2 sin A cos A

Answer 2

Step-by-step explanation:

Given :-

sin Ø , sin θ , cos Ø are the G.P.

cos² θ = k cos² [( π/4) + Ø].

To find :-

The value of k/8.

Solution :-

Given that

sin Ø , sin θ , cos Ø are the G.P.

We know that

If a, b, c are in the GP then b² = ac

So,

sin² θ = sin 2Ø × cos Ø

On multiplying with 2 both sides then

=> 2 sin² θ = 2 sin Ø × cos Ø

=> 2 sin² θ = sin 2Ø

Since, sin 2A = 2 sin A cos A

=> 2 sin² θ = sin 2Ø

=> 2( 1- cos² θ) = sin 2Ø

Since, sin² A + cos² A = 1

=> 2 - 2 cos² θ = sin 2Ø

=> 1 + 1 - 2 cos² θ = sin 2Ø

=> 1+ (1 - 2 cos² θ) = sin 2Ø

=> 1 -(2 cos² θ -1) = sin 2Ø

=> 1 - cos 2θ = sin 2Ø

Since, cos 2A = 2 cos² A -1

=> 1 - cos 2θ = - cos ( π/2 + 2Ø)

=> - cos 2θ = 1 - cos ( π/2 + 2Ø)

=> cos 2θ = -1 +cos ( π/2 + 2Ø)

=> cos 2θ = cos [( π/2) + 2Ø]+1

=> cos 2θ = cos 2[( π/4) + Ø]+1

=> cos 2θ =2 cos² [( π/4)+ Ø]-1+1

since, cos 2A = 2 cos² A -1

where , A = [(π/4)+Ø]

=> cos 2θ = 2 cos² [( π/4)+ Ø]

Given that

cos 2θ = k cos² [( π/4)+ Ø]

on comparing them

we have , k = 2

Now, The value of k/8 = 2/8 = 1/4

Answer :-

The value of k/8 = 1/4

Used formulae:-

→ If a, b, c are in the GP then b² = ac

→ sin² A + cos² A = 1

→ cos 2A = 2 cos² A -1

→ sin 2A = 2 sin A cos A

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