# Relating Trigonometric Functions

## Key Questions

• The reciprocal functions are as follows:

$\sin \left(a\right) \cdot \csc \left(a\right) = 1$

$\cos \left(a\right) \cdot \sec \left(a\right) = 1$

$\tan \left(a\right) \cdot \cot \left(a\right) = 1$

• It means to determine if the value of a trigonometric function is positive or negative; for example, since $\sin \left(\frac{3 \pi}{2}\right) = - 1 < 0$, its sign is negative, and since $\cos \left(- \frac{\pi}{3}\right) = \frac{1}{2} > 0$, its sign is positive.

I hope that this was helpful.

• The Pythagorean identity is:

color(red)(sin^2x+cos^2x=1

However, it does not have to apply to just sine and cosine.

To find the form of the Pythagorean identity with the other trigonometric identities, divide the original identity by sine and cosine.

SINE:

$\frac{{\sin}^{2} x + {\cos}^{2} x = 1}{\sin} ^ 2 x$

This gives:

${\sin}^{2} \frac{x}{\sin} ^ 2 x + {\cos}^{2} \frac{x}{\sin} ^ 2 x = \frac{1}{\sin} ^ 2 x$

Which equals

color(red)(1+cot^2x=csc^2x

To find the other identity:

COSINE:

$\frac{{\sin}^{2} x + {\cos}^{2} x = 1}{\cos} ^ 2 x$

This gives:

${\sin}^{2} \frac{x}{\cos} ^ 2 x + {\cos}^{2} \frac{x}{\cos} ^ 2 x = \frac{1}{\cos} ^ 2 x$

Which equals

color(red)(tan^2x+1=sec^2x

These identities can all be algebraically manipulated to prove many things:

$\left\{\begin{matrix}{\sin}^{2} x = 1 - {\cos}^{2} x \\ {\cos}^{2} x = 1 - {\sin}^{2} x\end{matrix}\right.$

$\left\{\begin{matrix}{\tan}^{2} x = {\sec}^{2} x - 1 \\ {\cot}^{2} x = {\csc}^{2} x - 1\end{matrix}\right.$

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