Question

sin 7 Theta = Cos theta​

Answer 1

Step-by-step explanation:

Sin7θ = Cosθ

=> Sin7θ = Sin ( π/2 - θ )

If we want to find general value --->

=> 7θ = π / 2 - θ

=> 7θ + θ = π / 2

=> 8θ = π / 2

=> θ = π / 16

If we want general value --->

Sin7θ = Sin( π / 2 - θ )

Sin7θ = Sin { nπ + ( -1 )ⁿ θ }

=> 7θ = nπ + ( -1 )ⁿ θ ...........................(1)

Let n = 2m is an even number

=> 7θ = 2mπ + ( -1 )²ᵐ θ

=> 7θ = 2mπ + ( 1 ) θ

=> 7θ - θ = 2mπ

=> 6θ = 2mπ

=> θ = 2mπ / 6

=> θ = mπ / 3

Let n = 2m + 1 is odd , so

=> 7θ = ( 2m + 1 )π + ( -1 )²ᵐ⁺¹ θ

=> 7θ = ( 2m + 1 ) π - θ

=> 7θ + θ = ( 2m + 1 ) π

=> 8θ = ( 2m + 1 ) π

=> θ = (2m + 1 ) π / 8

Answer 2

Answer:

Step-by-step explanation:

1) First take a look at this link which is a guide for DeMoivre's formula.

2) Using step 1 show that

sin(7x)=64sin(x)cos(x)^6−80sin(x)cos(x)^4+24sin(x)cos(x)^2−sin(x)

3) Replace cos2(x)=1−sin2(x) and obtain

sin(7x)=7sin(x)−56sin(x)^3+112sin(x)^5−64sin(x)^7

4) Divide by sin(x)

sin(7x)/sin(x)=7−56sin(x)^2+112sin(x)^4−64sin(x)^6