# 5.3 The other trigonometric functions  (Page 6/13)

 Page 6 / 13

Given an angle measure in radians, use a graphing utility/calculator to find the cosecant.

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

## Evaluating the secant using technology

Evaluate the cosecant of $\text{\hspace{0.17em}}\frac{5\pi }{7}.\text{\hspace{0.17em}}$

For a scientific calculator, enter information as follows:

$\mathrm{csc}\left(\frac{5\pi }{7}\right)\approx 1.279$

Evaluate the cotangent of $\text{\hspace{0.17em}}-\frac{\pi }{8}.\text{\hspace{0.17em}}$

$\approx -2.414$

Access these online resources for additional instruction and practice with other trigonometric functions.

## Key equations

 Tangent function $\mathrm{tan}\text{\hspace{0.17em}}t=\frac{\mathrm{sin}t}{\mathrm{cos}t}$ Secant function $\mathrm{sec}\text{\hspace{0.17em}}t=\frac{1}{\mathrm{cos}t}$ Cosecant function $\mathrm{csc}\text{\hspace{0.17em}}t=\frac{1}{\mathrm{sin}t}$ Cotangent function $\mathrm{cot}\text{\hspace{0.17em}}t=\frac{1}{\mathrm{tan}\text{\hspace{0.17em}}t}=\frac{\mathrm{cos}\text{\hspace{0.17em}}t}{\mathrm{sin}\text{\hspace{0.17em}}t}$

## Key concepts

• The tangent of an angle is the ratio of the y -value to the x -value of the corresponding point on the unit circle.
• The secant, cotangent, and cosecant are all reciprocals of other functions. The secant is the reciprocal of the cosine function, the cotangent is the reciprocal of the tangent function, and the cosecant is the reciprocal of the sine function.
• The six trigonometric functions can be found from a point on the unit circle. See [link] .
• Trigonometric functions can also be found from an angle. See [link] .
• Trigonometric functions of angles outside the first quadrant can be determined using reference angles. See [link] .
• A function is said to be even if $\text{\hspace{0.17em}}f\left(-x\right)=f\left(x\right)\text{\hspace{0.17em}}$ and odd if $\text{\hspace{0.17em}}f\left(-x\right)=-f\left(x\right).$
• Cosine and secant are even; sine, tangent, cosecant, and cotangent are odd.
• Even and odd properties can be used to evaluate trigonometric functions. See [link] .
• The Pythagorean Identity makes it possible to find a cosine from a sine or a sine from a cosine.
• Identities can be used to evaluate trigonometric functions. See [link] and [link] .
• Fundamental identities such as the Pythagorean Identity can be manipulated algebraically to produce new identities. See [link] .
• The trigonometric functions repeat at regular intervals.
• 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 smallest interval such that $\text{\hspace{0.17em}}f\left(x+P\right)=f\left(x\right)\text{\hspace{0.17em}}$ for any value of $\text{\hspace{0.17em}}x.$
• The values of trigonometric functions of special angles can be found by mathematical analysis.
• To evaluate trigonometric functions of other angles, we can use a calculator or computer software. See [link] .

## Verbal

On an interval of $\text{\hspace{0.17em}}\left[0,2\pi \right),$ can the sine and cosine values of a radian measure ever be equal? If so, where?

Yes, when the reference angle is $\text{\hspace{0.17em}}\frac{\pi }{4}\text{\hspace{0.17em}}$ and the terminal side of the angle is in quadrants I and III. Thus, at $\text{\hspace{0.17em}}x=\frac{\pi }{4},\frac{5\pi }{4},$ the sine and cosine values are equal.

What would you estimate the cosine of $\text{\hspace{0.17em}}\pi \text{\hspace{0.17em}}$ degrees to be? Explain your reasoning.

For any angle in quadrant II, if you knew the sine of the angle, how could you determine the cosine of the angle?

Substitute the sine of the angle in for $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ in the Pythagorean Theorem $\text{\hspace{0.17em}}{x}^{2}+{y}^{2}=1.\text{\hspace{0.17em}}$ Solve for $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ and take the negative solution.

Describe the secant function.

Tangent and cotangent have a period of $\text{\hspace{0.17em}}\pi .\text{\hspace{0.17em}}$ What does this tell us about the output of these functions?

The outputs of tangent and cotangent will repeat every $\text{\hspace{0.17em}}\pi \text{\hspace{0.17em}}$ units.

what is set?
a colony of bacteria is growing exponentially doubling in size every 100 minutes. how much minutes will it take for the colony of bacteria to triple in size
I got 300 minutes. is it right?
Patience
no. should be about 150 minutes.
Jason
It should be 158.5 minutes.
Mr
ok, thanks
Patience
100•3=300 300=50•2^x 6=2^x x=log_2(6) =2.5849625 so, 300=50•2^2.5849625 and, so, the # of bacteria will double every (100•2.5849625) = 258.49625 minutes
Thomas
what is the importance knowing the graph of circular functions?
can get some help basic precalculus
What do you need help with?
Andrew
how to convert general to standard form with not perfect trinomial
can get some help inverse function
ismail
Rectangle coordinate
how to find for x
it depends on the equation
Robert
yeah, it does. why do we attempt to gain all of them one side or the other?
Melissa
whats a domain
The domain of a function is the set of all input on which the function is defined. For example all real numbers are the Domain of any Polynomial function.
Spiro
Spiro; thanks for putting it out there like that, 😁
Melissa
foci (–7,–17) and (–7,17), the absolute value of the differenceof the distances of any point from the foci is 24.
difference between calculus and pre calculus?
give me an example of a problem so that I can practice answering
x³+y³+z³=42
Robert
dont forget the cube in each variable ;)
Robert
of she solves that, well ... then she has a lot of computational force under her command ....
Walter
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