Question

Which of the following are solutions to the equation sinx cos(pi/7)-sin(pi/7)cosx=sqrt(2)/2

check all that apply
A. 7pi/4+pi/4+2npi
B. 5pi/4+pi/7+2npi
C. pi/4+pi/7+2npi
D. 3pi/4+pi/7+2npi

Answer 1

The correct answer is:

Option: D

D. 3pi/4+pi/7+2npi

Explanation:

We know that:

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

Here we are given:

`\sin x\cos ((\pi)/(7))-\sin ((\pi)/(7))\cos x=(√(2))/(2)=(1)/(√(2))`

so, this quantity as by the above formula will be equal to :

`\sin (x-(\pi)/(7))=(1)/(√(2))`

Now, we will check which options or which value of x will hold the following equation true.

Option: D is the correct answer.

i.e. when we put:

`x=(3\pi)/(4)+(\pi)/(7)+2n\pi`

we get:

`\sin ((3\pi)/(4)+(\pi)/(7)+2n\pi-(\pi)/(7))=(1)/(√(2))\\\\\sin (2n\pi+(3\pi)/(4))=(1)/(√(2))\\\\\sin ((3\pi)/(4))=(1)/(√(2))\\\\\sin (\pi-(\pi)/(4))=(1)/(√(2))\\\\\sin ((\pi)/(4))=(1)/(√(2))\\\\(1)/(√(2))=(1)/(√(2))`

Answer 2

The solution would be like this for this specific problem:

sin ( x - ( pi / 7 ) ) = - sqrt ( 2 ) / 2
x - ( pi / 7 ) = - pi / 4 + 2n*pi or x - ( pi / 7 ) = (5pi / 4 ) + 2n*pi
x = ( pi / 7 ) - ( pi / 4 ) + 2n*pi or x = ( 5pi / 4 ) + ( pi / 7 ) + 2n*pi
x = ( - 3pi / 28 ) + 2n*pi

I am hoping that this answer has satisfied your query and it will be able to help you in your endeavor, and if you would like, feel free to ask another question.