Question

Solve this Equation:

sin(x+pi/4)-sin(x-pi/4)=1

I do not just want the answer. I want this explained to me so I understand it. Thank you!

Answer 1

The formula that I would use is the difference formula for sine, right? The difference formula for sin is: sin(a-b)= sina cosb- cosa sinb

Answer 2

@mathmate Please help me! I will award metal???

Answer 3

@hick4life

Answer 4

ok i got cha

Answer 5

You have the right approach, can you continue?
@HaileyJCaroen

Answer 6

open up" each of the terms, using the definition

Answer 7

Well what I have is
=sin(x+pi/4)-sin(x-pi/4)
=-2sin(x+pi/4-x+pi/4) cos(x+pi/4-X+pi/4) /2

I don't know if that is right.

Answer 8

well I know that a=x and pi/4=b

Answer 9

So I guess I am supposed to plug in a and b into the difference formula for sine.
So that would be:
sin(x-pi/4)=sin(x) cos(pi/4)- cos(x) sin(pi/4)

Answer 10

exactly, do the same for both sin(x+pi/4) and sin(x-pi/4), when you add the terms, something magical will happen! :)

Answer 11

What do you mean?

Answer 12

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

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

therefore,

sin(x+pi/4)+sin(x-pi/4) = sin(x)cos(pi/4)+sin(pi/4)cos(x) + sin(x)cos(pi/4)-sin(pi/4)cos(x)
= 2*sin(x) cos(pi/4)

Now, cos(pi/4)=cos(45 degrees) = 1/sqrt(2).

Therefore

sin(x+pi/4)+sin(x-pi/4)=2*(1/sqrt(2))*... = sqrt(2) * sin(x)

and finally, we make this equal to 1:

sin(x+pi/4) +sin(x-pi/4) = sqrt(2) sin(x) = 1

therefore the x's that solve this will be:

sin(x)=1/sqrt(2) --> x=pi/4, 3pi/4, 9pi/4, 11pi/4, 15pi/4, 17pi/4, and also -7pi/4, -5pi/4, -13pi/4, -15pi/4...etc

in general:

x= pi/4 (plus minus) 2*pi

and

x = - pi/4 (plus minus) 2*pi

the "plus minus" 2*pi is considering any number of cycles you want to make around the unit circle.

Answer 13

Oh I get it. You use both difference formulas and add them.

Answer 14

Thank you both for helping me! I have a similar question I need help with.

Answer 15

no problem