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

 Page 3 / 13

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.

#### Questions & Answers

how can are find the domain and range of a relations
A cell phone company offers two plans for minutes. Plan A: $15 per month and$2 for every 300 texts. Plan B: $25 per month and$0.50 for every 100 texts. How many texts would you need to send per month for plan B to save you money?
6000
Robert
more than 6000
Robert
can I see the picture
How would you find if a radical function is one to one?
how to understand calculus?
with doing calculus
SLIMANE
Thanks po.
Jenica
Hey I am new to precalculus, and wanted clarification please on what sine is as I am floored by the terms in this app? I don't mean to sound stupid but I have only completed up to college algebra.
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
Jenica
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).
Natalie
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
What is domain
johnphilip
the standard equation of the ellipse that has vertices (0,-4)&(0,4) and foci (0, -15)&(0,15) it's standard equation is x^2 + y^2/16 =1 tell my why is it only x^2? why is there no a^2?
what is foci?
This term is plural for a focus, it is used for conic sections. For more detail or other math questions. I recommend researching on "Khan academy" or watching "The Organic Chemistry Tutor" YouTube channel.
Chris
how to determine the vertex,focus,directrix and axis of symmetry of the parabola by equations
i want to sure my answer of the exercise
what is the diameter of(x-2)²+(y-3)²=25
how to solve the Identity ?
what type of identity
Jeffrey
Confunction Identity
Barcenas
how to solve the sums
meena
hello guys
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?
Shakeena
32.243
Kenard
how solve standard form of polar
what is a complex number used for?
It's just like any other number. The important thing to know is that they exist and can be used in computations like any number.
Steve
I would like to add that they are used in AC signal analysis for one thing
Scott
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