# Thread: Showing how pi/2+theta = cos using unit circle.

1. ## Showing how pi/2+theta = cos using unit circle.

Hey all,

Been stuck with my classwork lately, I can not visualize why pi/2+theta = cos theta. This is similar to my last post except i was able to understand it when its in the first quadrant.

Thanks.

2. $\sin{\left(\frac{\pi}{2}+\theta\right)}=\cos{\thet a}$

You mean this correct?

3. dwsmith, yes.

4. Originally Posted by Oiler
I can not visualize why pi/2+theta = cos theta. Thanks.
Draw $\sin{\left(\frac{\pi}{2}+\theta\right)}$ and $\cos{\theta}$ separately on the same set of axes.

What do you notice?

5. The angles are symmetrical at pi/2 ?

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6. Originally Posted by Oiler
The angles are symmetrical at pi/2 ?

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At $\theta = \frac{\pi}{2}$ they are the same.

Did you plot them together yet?

7. This is all i can imagine the angle to be xx - Slimber.com: Drawing and Painting Online

8. If you put the $\theta$ between the vertical axis instead, then where you currently have $\theta$ will be $\pi-(\theta+\frac{\pi}2)$ and we know $sin ({\pi-(\theta+\frac{\pi}2})) = sin( \theta+\frac{\pi}2)$ and by alternate angles, the other angle in the triangle will be $\theta$.
see diagram

$cos \theta = \frac{y}1$
$sin (\pi-(\theta+\frac{\pi}2}))=\frac{y}1$
hence $sin (\theta+\frac{\pi}2)=cos \theta$

Attachment 20466

9. Thanks jacs. i can clearly see this now.

10. This in theory should also hold true to cos(pi/2+theta)=sinx ?..

11. Originally Posted by Oiler
This in theory should also hold true to cos(pi/2+theta)=sinx ?..
Nope, it is $\cos \left( \theta -\frac{\pi}{2}\right) = \sin \theta$

12. hmm for cos(pi/2+theta) i get cos(pi/2+theta) = sin(theta), I am guessing it should be -sin(theta)

13. Originally Posted by Oiler
hmm for cos(pi/2+theta) i get cos(pi/2+theta) = sin(theta), I am guessing it should be -sin(theta)
Looking at the unit circle, adding pi/2 to an angle in the first quadrant gives and angle in the second quadrant where the "x" coordinate is negative. You should also be able to see that the vertical and horizontal sides of the right triangles formed (|x| and |y|) are swapped. That is (x, y), in the first quadrant, with both x and y positive, is rotated to (-y, x).

Since, on the unit circle, the first coordinate is $cos(\theta)$ and the second coordinate is $sin(theta)$, $cos(\theta+ \pi/2)= -y= -sin(\theta)$.

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