# 5.3 The other trigonometric functions  (Page 5/13)

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Other functions can also be periodic. For example, the lengths of months repeat every four years. If $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ represents the length time, measured in years, and $\text{\hspace{0.17em}}f\left(x\right)\text{\hspace{0.17em}}$ represents the number of days in February, then $\text{\hspace{0.17em}}f\left(x+4\right)=f\left(x\right).\text{\hspace{0.17em}}$ This pattern repeats over and over through time. In other words, every four years, February is guaranteed to have the same number of days as it did 4 years earlier. The positive number 4 is the smallest positive number that satisfies this condition and is called the period. A period is the shortest interval over which a function completes one full cycle—in this example, the period is 4 and represents the time it takes for us to be certain February has the same number of days.

## Period of a function

The period     $\text{\hspace{0.17em}}P\text{\hspace{0.17em}}$ of a repeating function $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ is the number representing the interval such that $\text{\hspace{0.17em}}f\left(x+P\right)=f\left(x\right)\text{\hspace{0.17em}}$ for any value of $\text{\hspace{0.17em}}x.\text{\hspace{0.17em}}$

The period of the cosine, sine, secant, and cosecant functions is $\text{\hspace{0.17em}}2\pi .\text{\hspace{0.17em}}$

The period of the tangent and cotangent functions is $\text{\hspace{0.17em}}\pi .$

## Finding the values of trigonometric functions

Find the values of the six trigonometric functions of angle $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ based on [link] .

$\begin{array}{l}\mathrm{sin}\text{\hspace{0.17em}}t=y=-\frac{\sqrt{3}}{2}\\ \mathrm{cos}\text{\hspace{0.17em}}t=x=-\frac{1}{2}\\ \mathrm{tan}\text{\hspace{0.17em}}t=\frac{\mathrm{sin}t}{\mathrm{cos}t}=\frac{-\frac{\sqrt{3}}{2}}{-\frac{1}{2}}=\sqrt{3}\\ \mathrm{sec}\text{\hspace{0.17em}}t=\frac{1}{\mathrm{cos}t}=\frac{1}{-\frac{1}{2}}=-2\\ \mathrm{csc}\text{\hspace{0.17em}}t=\frac{1}{\mathrm{sin}t}=\frac{1}{-\frac{\sqrt{3}}{2}}=-\frac{2\sqrt{3}}{3}\\ \mathrm{cot}\text{\hspace{0.17em}}t=\frac{1}{\mathrm{tan}t}=\frac{1}{\sqrt{3}}=\frac{\sqrt{3}}{3}\end{array}$

Find the values of the six trigonometric functions of angle $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ based on [link] .

$\begin{array}{l}\mathrm{sin}\text{\hspace{0.17em}}t=-1,\mathrm{cos}\text{\hspace{0.17em}}t=0,\mathrm{tan}\text{\hspace{0.17em}}t=\text{Undefined}\\ \mathrm{sec}\text{\hspace{0.17em}}t=\text{\hspace{0.17em}Undefined,}\mathrm{csc}\text{\hspace{0.17em}}t=-1,\mathrm{cot}\text{\hspace{0.17em}}t=0\end{array}$

## Finding the value of trigonometric functions

If $\text{\hspace{0.17em}}\mathrm{sin}\left(t\right)=-\frac{\sqrt{3}}{2}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\text{cos}\left(t\right)=\frac{1}{2},$ find

$\begin{array}{l}\begin{array}{l}\\ \mathrm{sec}\text{\hspace{0.17em}}t=\frac{1}{\mathrm{cos}\text{\hspace{0.17em}}t}=\frac{1}{\frac{1}{2}}=2\end{array}\hfill \\ \mathrm{csc}\text{\hspace{0.17em}}t=\frac{1}{\mathrm{sin}\text{\hspace{0.17em}}t}=\frac{1}{-\frac{\sqrt{3}}{2}}-\frac{2\sqrt{3}}{3}\hfill \\ \mathrm{tan}\text{\hspace{0.17em}}t=\frac{\mathrm{sin}\text{\hspace{0.17em}}t}{\mathrm{cos}\text{\hspace{0.17em}}t}=\frac{-\frac{\sqrt{3}}{2}}{\frac{1}{2}}=-\sqrt{3}\hfill \\ \mathrm{cot}\text{\hspace{0.17em}}t=\frac{1}{\mathrm{tan}\text{\hspace{0.17em}}t}=\frac{1}{-\sqrt{3}}=-\frac{\sqrt{3}}{3}\hfill \end{array}$

If $\text{\hspace{0.17em}}\mathrm{sin}\left(t\right)=\frac{\sqrt{2}}{2}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\mathrm{cos}\left(t\right)=\frac{\sqrt{2}}{2},$ find

$\mathrm{sec}\text{\hspace{0.17em}}t=\sqrt{2},\mathrm{csc}\text{\hspace{0.17em}}t=\sqrt{2},\mathrm{tan}\text{\hspace{0.17em}}t=1,\mathrm{cot}\text{\hspace{0.17em}}t=1$

## Evaluating trigonometric functions with a calculator

We have learned how to evaluate the six trigonometric functions for the common first-quadrant angles and to use them as reference angles for angles in other quadrants. To evaluate trigonometric functions of other angles, we use a scientific or graphing calculator or computer software. If the calculator has a degree mode and a radian mode, confirm the correct mode is chosen before making a calculation.

Evaluating a tangent function with a scientific calculator as opposed to a graphing calculator or computer algebra system is like evaluating a sine or cosine: Enter the value and press the TAN key. For the reciprocal functions, there may not be any dedicated keys that say CSC, SEC, or COT. In that case, the function must be evaluated as the reciprocal of a sine, cosine, or tangent.

If we need to work with degrees and our calculator or software does not have a degree mode, we can enter the degrees multiplied by the conversion factor $\text{\hspace{0.17em}}\frac{\pi }{180}\text{\hspace{0.17em}}$ to convert the degrees to radians. To find the secant of $\text{\hspace{0.17em}}30°,$ we could press

or

Given an angle measure in radians, use a scientific calculator to find the cosecant.

1. If the calculator has degree mode and radian mode, set it to radian mode.
2. Enter:
3. Enter the value of the angle inside parentheses.
4. Press the SIN key.
5. Press the = key.

#### Questions & Answers

what is a function?
I want to learn about the law of exponent
explain this
what is functions?
A mathematical relation such that every input has only one out.
Spiro
yes..it is a relationo of orders pairs of sets one or more input that leads to a exactly one output.
Mubita
Is a rule that assigns to each element X in a set A exactly one element, called F(x), in a set B.
RichieRich
If the plane intersects the cone (either above or below) horizontally, what figure will be created?
can you not take the square root of a negative number
No because a negative times a negative is a positive. No matter what you do you can never multiply the same number by itself and end with a negative
lurverkitten
Actually you can. you get what's called an Imaginary number denoted by i which is represented on the complex plane. The reply above would be correct if we were still confined to the "real" number line.
Liam
Suppose P= {-3,1,3} Q={-3,-2-1} and R= {-2,2,3}.what is the intersection
can I get some pretty basic questions
In what way does set notation relate to function notation
Ama
is precalculus needed to take caculus
It depends on what you already know. Just test yourself with some precalculus questions. If you find them easy, you're good to go.
Spiro
the solution doesn't seem right for this problem
what is the domain of f(x)=x-4/x^2-2x-15 then
x is different from -5&3
Seid
All real x except 5 and - 3
Spiro
***youtu.be/ESxOXfh2Poc
Loree
how to prroved cos⁴x-sin⁴x= cos²x-sin²x are equal
Don't think that you can.
Elliott
By using some imaginary no.
Tanmay
how do you provided cos⁴x-sin⁴x = cos²x-sin²x are equal
What are the question marks for?
Elliott
Someone should please solve it for me Add 2over ×+3 +y-4 over 5 simplify (×+a)with square root of two -×root 2 all over a multiply 1over ×-y{(×-y)(×+y)} over ×y
For the first question, I got (3y-2)/15 Second one, I got Root 2 Third one, I got 1/(y to the fourth power) I dont if it's right cause I can barely understand the question.
Is under distribute property, inverse function, algebra and addition and multiplication function; so is a combined question
Abena
find the equation of the line if m=3, and b=-2
graph the following linear equation using intercepts method. 2x+y=4
Ashley
how
Wargod
what?
John
ok, one moment
UriEl
how do I post your graph for you?
UriEl
it won't let me send an image?
UriEl
also for the first one... y=mx+b so.... y=3x-2
UriEl
y=mx+b you were already given the 'm' and 'b'. so.. y=3x-2
Tommy
Please were did you get y=mx+b from
Abena
y=mx+b is the formula of a straight line. where m = the slope & b = where the line crosses the y-axis. In this case, being that the "m" and "b", are given, all you have to do is plug them into the formula to complete the equation.
Tommy
thanks Tommy
Nimo
0=3x-2 2=3x x=3/2 then . y=3/2X-2 I think
Given
co ordinates for x x=0,(-2,0) x=1,(1,1) x=2,(2,4)
neil