# Thread: why sin 90+theta is not equal to sin 180-theta?

1. ## why sin 90+theta is not equal to sin 180-theta?

Given the formula sin 90+theta = cos theta and sin 180 - theta = sin theta
why sin 90+theta is not equal to sin 180-theta since both of them belong to 2nd quadrant in the unit circle.

2. ## Re: why sin 90+theta is not equal to sin 180-theta?

Originally Posted by hisajesh
Given the formula sin 90+theta = cos theta and sin 180 - theta = sin theta
why sin 90+theta is not equal to sin 180-theta since both of them belong to 2nd quadrant in the unit circle.
You can do this with diagrams, but it's simpler (for me anyway) to do it according the sum of angle formulas. In general $\displaystyle sin( A + B ) = sin(A)~cos(B) + sin(B)~cos(A)$

$\displaystyle sin(90 + \theta) = sin(90)~cos( \theta ) + sin( \theta )~cos(90) = cos( \theta ) + 0 = cos(\theta)$

$\displaystyle sin(180 - \theta ) = sin(180)~cos( \theta ) - sin( \theta )~cos(180) = 0 + sin( \theta ) = sin( \theta )$ (Watch the double negative in the last step.)

-Dan

3. ## Re: why sin 90+theta is not equal to sin 180-theta?

see attached diagram ...

4. ## Re: why sin 90+theta is not equal to sin 180-theta?

They both belong to the second quadrant but 90+ 30= 120 degrees is not equal to 180- 30= 150 degrees so why should they be equal?

5. ## Re: why sin 90+theta is not equal to sin 180-theta?

@Skeeter: I need to ask, from your diagram, why the perependicular is across x axis why should not be it across y - axis. Can we compute the ratio that way?

6. ## Re: why sin 90+theta is not equal to sin 180-theta?

sine of an angle in standard position is defined as the ratio y/r

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### sin(90-theta) = cos(theta)

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