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.
This is all i can imagine the angle to be xx - Slimber.com: Drawing and Painting Online
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
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 [itex]cos(\theta)[/itex] and the second coordinate is [itex]sin(theta)[/itex], [itex]cos(\theta+ \pi/2)= -y= -sin(\theta)[/itex].