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# Relating Trigonometric Functions

## Key Questions

• The reciprocal functions are as follows:

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

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

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

As below

#### Explanation:

Quotient Identities. There are two quotient identities that can be used in right triangle trigonometry.

A quotient identity defines the relations for tangent and cotangent in terms of sine and cosine. ...

.

Remember that the difference between an equation and an identity is that an identity will be true for ALL values.

• 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:

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

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

• This key question hasn't been answered yet.

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