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

• Jan 16th 2011, 08:06 PM
Oiler
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.
• Jan 16th 2011, 08:10 PM
dwsmith
$\displaystyle \sin{\left(\frac{\pi}{2}+\theta\right)}=\cos{\thet a}$

You mean this correct?
• Jan 16th 2011, 08:27 PM
Oiler
dwsmith, yes.
• Jan 16th 2011, 08:29 PM
pickslides
Quote:

Originally Posted by Oiler
I can not visualize why pi/2+theta = cos theta. Thanks.

Draw $\displaystyle \sin{\left(\frac{\pi}{2}+\theta\right)}$ and $\displaystyle \cos{\theta}$ separately on the same set of axes.

What do you notice?
• Jan 16th 2011, 08:35 PM
Oiler
The angles are symmetrical at pi/2 ?

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• Jan 16th 2011, 09:28 PM
pickslides
Quote:

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

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

Did you plot them together yet?
• Jan 16th 2011, 09:47 PM
Oiler
This is all i can imagine the angle to be xx - Slimber.com: Drawing and Painting Online
• Jan 16th 2011, 10:16 PM
jacs
If you put the $\displaystyle \theta$ between the vertical axis instead, then where you currently have $\displaystyle \theta$ will be $\displaystyle \pi-(\theta+\frac{\pi}2)$ and we know $\displaystyle sin ({\pi-(\theta+\frac{\pi}2})) = sin( \theta+\frac{\pi}2)$ and by alternate angles, the other angle in the triangle will be $\displaystyle \theta$.
see diagram

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

Attachment 20466
• Jan 16th 2011, 10:23 PM
Oiler
Thanks jacs. i can clearly see this now.
• Jan 16th 2011, 10:29 PM
Oiler
This in theory should also hold true to cos(pi/2+theta)=sinx ?..
• Jan 16th 2011, 10:33 PM
pickslides
Quote:

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

Nope, it is $\displaystyle \cos \left( \theta -\frac{\pi}{2}\right) = \sin \theta$
• Jan 16th 2011, 10:48 PM
Oiler
hmm for cos(pi/2+theta) i get cos(pi/2+theta) = sin(theta), I am guessing it should be -sin(theta)
• Jan 17th 2011, 11:43 AM
HallsofIvy
Quote:

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 $sin(theta)$, $cos(\theta+ \pi/2)= -y= -sin(\theta)$.