Co-Function Identities

Derivation

A trig identity based on Complementary Angles If we have a right angle triangle, the angles must add up to 180 Since we already used a 90 degree angle, we are left with theta. the complementary angle finishes it. The sin of theta would be using these 2 lengths The cos of theta’s complement would be using these 2 lengths as well. angles that are complements of each other will have the same sine → cosine and same cosine → sine. Such is visible in the unit circle as well. a few trig ratios in the same quadrant have the same values, just in reverse order. The sin of $90^o$ is the cos of $0^o$ The sin of $60^o$ is the cos of $30^o$

Thus, we can say the $\sin \theta =\cos (90-\theta )$

Identities

To find identities, one must be willing to draw the quadrants, and predict. You have 3 possible ways the trig function can look: $function(\frac{\pi }{2}+x)$ $function(\frac{\pi }{2}-x)$ $function(x-\frac{\pi }{2})$ You can do all 3 of them, or one of them

If x is provided, use x. If x is not provided, assume x = 30, then figure out.

lets do it for one of them. $function(\frac{\pi }{2}+x)$

1. Draw out the list of functions like such:
2. Make each function equal its opposite version
3. As x was not provided, we must assume x = 30. Now fill in the chart as where the transposed angle ended up(in the second quadrant).