# 5.3 The other trigonometric functions  (Page 3/13)

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Use reference angles to find all six trigonometric functions of $\text{\hspace{0.17em}}-\frac{7\pi }{4}.\text{\hspace{0.17em}}$

$\mathrm{sin}\left(\frac{-7\pi }{4}\right)=\frac{\sqrt{2}}{2},\mathrm{cos}\left(\frac{-7\pi }{4}\right)=\frac{\sqrt{2}}{2},\mathrm{tan}\left(\frac{-7\pi }{4}\right)=1,$
$\mathrm{sec}\left(\frac{-7\pi }{4}\right)=\sqrt{2},\mathrm{csc}\left(\frac{-7\pi }{4}\right)=\sqrt{2},\mathrm{cot}\left(\frac{-7\pi }{4}\right)=1$

## Using even and odd trigonometric functions

To be able to use our six trigonometric functions freely with both positive and negative angle inputs, we should examine how each function treats a negative input. As it turns out, there is an important difference among the functions in this regard.

Consider the function $\text{\hspace{0.17em}}f\left(x\right)={x}^{2},$ shown in [link] . The graph of the function is symmetrical about the y -axis. All along the curve, any two points with opposite x -values have the same function value. This matches the result of calculation: $\text{\hspace{0.17em}}{\left(4\right)}^{2}={\left(-4\right)}^{2},$ ${\left(-5\right)}^{2}={\left(5\right)}^{2},$ and so on. So $\text{\hspace{0.17em}}f\left(x\right)={x}^{2}\text{\hspace{0.17em}}$ is an even function    , a function such that two inputs that are opposites have the same output. That means $\text{\hspace{0.17em}}f\left(-x\right)=f\left(x\right).\text{\hspace{0.17em}}$

Now consider the function $\text{\hspace{0.17em}}f\left(x\right)={x}^{3},$ shown in [link] . The graph is not symmetrical about the y -axis. All along the graph, any two points with opposite x -values also have opposite y -values. So $\text{\hspace{0.17em}}f\left(x\right)={x}^{3}\text{\hspace{0.17em}}$ is an odd function    , one such that two inputs that are opposites have outputs that are also opposites. That means $\text{\hspace{0.17em}}f\left(-x\right)=-f\left(x\right).\text{\hspace{0.17em}}$

We can test whether a trigonometric function is even or odd by drawing a unit circle    with a positive and a negative angle, as in [link] . The sine of the positive angle is $\text{\hspace{0.17em}}y.\text{\hspace{0.17em}}$ The sine of the negative angle is − y . The sine function    , then, is an odd function. We can test each of the six trigonometric functions in this fashion. The results are shown in [link] .

 $\begin{array}{l}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}t=y\hfill \\ \mathrm{sin}\left(-t\right)=-y\hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}t\ne \mathrm{sin}\left(-t\right)\hfill \end{array}$ $\begin{array}{l}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{cos}\text{\hspace{0.17em}}t=x\hfill \\ \mathrm{cos}\left(-t\right)=x\hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}t=\mathrm{cos}\left(-t\right)\hfill \end{array}$ $\begin{array}{l}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{tan}\left(t\right)=\frac{y}{x}\hfill \\ \text{\hspace{0.17em}}\mathrm{tan}\left(-t\right)=-\frac{y}{x}\hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{tan}\text{\hspace{0.17em}}t\ne \mathrm{tan}\left(-t\right)\hfill \end{array}$ $\begin{array}{l}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{sec}\text{\hspace{0.17em}}t=\frac{1}{x}\hfill \\ \mathrm{sec}\left(-t\right)=\frac{1}{x}\hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{sec}\text{\hspace{0.17em}}t=\mathrm{sec}\left(-t\right)\hfill \end{array}$ $\begin{array}{l}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{csc}\text{\hspace{0.17em}}t=\frac{1}{y}\hfill \\ \text{\hspace{0.17em}}\mathrm{csc}\left(-t\right)=\frac{1}{-y}\hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{csc}\text{\hspace{0.17em}}t\ne \mathrm{csc}\left(-t\right)\hfill \end{array}$ $\begin{array}{l}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{cot}\text{\hspace{0.17em}}t=\frac{x}{y}\hfill \\ \text{\hspace{0.17em}}\mathrm{cot}\left(-t\right)=\frac{x}{-y}\hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{cot}\text{\hspace{0.17em}}t\ne cot\left(-t\right)\hfill \end{array}$

## Even and odd trigonometric functions

An even function    is one in which $\text{\hspace{0.17em}}f\left(-x\right)=f\left(x\right).$

An odd function    is one in which $\text{\hspace{0.17em}}f\left(-x\right)=-f\left(x\right).$

Cosine and secant are even:

$\begin{array}{l}\mathrm{cos}\left(-t\right)=\text{cos}\text{\hspace{0.17em}}t\hfill \\ \mathrm{sec}\left(-t\right)=\mathrm{sec}\text{\hspace{0.17em}}t\hfill \end{array}$

Sine, tangent, cosecant, and cotangent are odd:

$\begin{array}{l}\mathrm{sin}\left(-t\right)=-\mathrm{sin}\text{\hspace{0.17em}}t\hfill \\ \mathrm{tan}\left(-t\right)=-\mathrm{tan}\text{\hspace{0.17em}}t\hfill \\ \mathrm{csc}\left(-t\right)=-\mathrm{csc}\text{\hspace{0.17em}}t\hfill \\ \mathrm{cot}\left(-t\right)=-\mathrm{cot}\text{\hspace{0.17em}}t\hfill \end{array}$

## Using even and odd properties of trigonometric functions

If the secant of angle $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ is 2, what is the secant of $\text{\hspace{0.17em}}-t?\text{\hspace{0.17em}}$

Secant is an even function. The secant of an angle is the same as the secant of its opposite. So if the secant of angle t is 2, the secant of $\text{\hspace{0.17em}}-t\text{\hspace{0.17em}}$ is also 2.

If the cotangent of angle $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ is $\text{\hspace{0.17em}}\sqrt{3},$ what is the cotangent of $\text{\hspace{0.17em}}-t?\text{\hspace{0.17em}}$

$\text{\hspace{0.17em}}-\sqrt{3}\text{\hspace{0.17em}}$

## Recognizing and using fundamental identities

We have explored a number of properties of trigonometric functions. Now, we can take the relationships a step further, and derive some fundamental identities. Identities are statements that are true for all values of the input on which they are defined. Usually, identities can be derived from definitions and relationships we already know. For example, the Pythagorean Identity we learned earlier was derived from the Pythagorean Theorem and the definitions of sine and cosine.

how can are find the domain and range of a relations
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6000
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more than 6000
Robert
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SLIMANE
Thanks po.
Jenica
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I don't know if you are looking for a deeper answer or not, but the sine of an angle in a right triangle is the length of the opposite side to the angle in question divided by the length of the hypotenuse of said triangle.
Marco
can you give me sir tips to quickly understand precalculus. Im new too in that topic. Thanks
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if you remember sine, cosine, and tangent from geometry, all the relationships are the same but they use x y and r instead (x is adjacent, y is opposite, and r is hypotenuse).
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it is better to use unit circle than triangle .triangle is only used for acute angles but you can begin with. Download any application named"unit circle" you find in it all you need. unit circle is a circle centred at origine (0;0) with radius r= 1.
SLIMANE
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johnphilip
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meena
For each year t, the population of a forest of trees is represented by the function A(t) = 117(1.029)t. In a neighboring forest, the population of the same type of tree is represented by the function B(t) = 86(1.025)t.
by how many trees did forest "A" have a greater number?
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I would like to add that they are used in AC signal analysis for one thing
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Good call Scott. Also radar signals I believe.
Steve
They are used in any profession where the phase of a waveform has to be accounted for in the calculations. Imagine two electrical signals in a wire that are out of phase by 90°. At some times they will interfere constructively, others destructively. Complex numbers simplify those equations
Tim