# 5.3 The other trigonometric functions  (Page 4/13)

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## Fundamental identities

We can derive some useful identities    from the six trigonometric functions. The other four trigonometric functions can be related back to the sine and cosine functions using these basic relationships:

$\text{tan}\text{\hspace{0.17em}}t=\frac{\mathrm{sin}\text{\hspace{0.17em}}t}{\mathrm{cos}\text{\hspace{0.17em}}t}$
$\mathrm{sec}\text{\hspace{0.17em}}t=\frac{1}{\mathrm{cos}\text{\hspace{0.17em}}t}$
$\mathrm{csc}\text{\hspace{0.17em}}t=\frac{1}{\mathrm{sin}\text{\hspace{0.17em}}t}$
$\text{cot}\text{\hspace{0.17em}}t=\frac{1}{\text{tan}\text{\hspace{0.17em}}t}=\frac{\text{cos}\text{\hspace{0.17em}}t}{\text{sin}\text{\hspace{0.17em}}t}$

## Using identities to evaluate trigonometric functions

1. Given $\mathrm{sin}\left(45°\right)=\frac{\sqrt{2}}{2},\mathrm{cos}\left(45°\right)=\frac{\sqrt{2}}{2},$ evaluate $\mathrm{tan}\left(45°\right).$
2. Given $\text{\hspace{0.17em}}\mathrm{sin}\left(\frac{5\pi }{6}\right)=\frac{1}{2},\text{cos}\left(\frac{5\pi }{6}\right)=-\frac{\sqrt{3}}{2},\text{evaluate}\text{\hspace{0.17em}}\mathrm{sec}\left(\frac{5\pi }{6}\right).$

Because we know the sine and cosine values for these angles, we can use identities to evaluate the other functions.

1. $\begin{array}{l}\mathrm{tan}\left(45°\right)=\frac{\mathrm{sin}\left(45°\right)}{\mathrm{cos}\left(45°\right)}\hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}}\hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=1\hfill \end{array}$
2. $\begin{array}{l}\mathrm{sec}\left(\frac{5\pi }{6}\right)=\frac{1}{\mathrm{cos}\left(\frac{5\pi }{6}\right)}\hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\frac{1}{-\frac{\sqrt{3}}{2}}\hfill \\ \begin{array}{l}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\frac{-2\sqrt{3}}{1}\hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\frac{-2}{\sqrt{3}}\hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=-\frac{2\sqrt{3}}{3}\hfill \end{array}\hfill \end{array}$

Evaluate $\text{\hspace{0.17em}}\text{csc}\left(\frac{7\pi }{6}\right).$

$-2$

## Using identities to simplify trigonometric expressions

Simplify $\frac{\mathrm{sec}\text{\hspace{0.17em}}t}{\mathrm{tan}\text{\hspace{0.17em}}t}.$

We can simplify this by rewriting both functions in terms of sine and cosine.

By showing that $\text{\hspace{0.17em}}\frac{\mathrm{sec}\text{\hspace{0.17em}}t}{\mathrm{tan}\text{\hspace{0.17em}}t}\text{\hspace{0.17em}}$ can be simplified to $\text{\hspace{0.17em}}\mathrm{csc}\text{\hspace{0.17em}}t,$ we have, in fact, established a new identity.

$\frac{\mathrm{sec}\text{\hspace{0.17em}}t}{\mathrm{tan}\text{\hspace{0.17em}}t}=\mathrm{csc}\text{\hspace{0.17em}}t$

Simplify $\text{\hspace{0.17em}}\mathrm{tan}\text{\hspace{0.17em}}t\left(\mathrm{cos}\text{\hspace{0.17em}}t\right).$

$\mathrm{sin}\text{\hspace{0.17em}}t$

## Alternate forms of the pythagorean identity

We can use these fundamental identities to derive alternative forms of the Pythagorean Identity    , $\text{\hspace{0.17em}}{\mathrm{cos}}^{2}t+{\mathrm{sin}}^{2}t=1.\text{\hspace{0.17em}}$ One form is obtained by dividing both sides by $\text{\hspace{0.17em}}{\mathrm{cos}}^{2}t:$

$\begin{array}{c}\frac{{\mathrm{cos}}^{2}t}{{\mathrm{cos}}^{2}t}+\frac{{\mathrm{sin}}^{2}t}{{\mathrm{cos}}^{2}t}=\frac{1}{{\mathrm{cos}}^{2}t}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}1+{\mathrm{tan}}^{2}t={\mathrm{sec}}^{2}t\end{array}$

The other form is obtained by dividing both sides by $\text{\hspace{0.17em}}{\mathrm{sin}}^{2}t:$

$\begin{array}{c}\frac{{\mathrm{cos}}^{2}t}{{\mathrm{sin}}^{2}t}+\frac{{\mathrm{sin}}^{2}t}{{\mathrm{sin}}^{2}t}=\frac{1}{{\mathrm{sin}}^{2}t}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\mathrm{cot}}^{2}t+1={\mathrm{csc}}^{2}t\end{array}$

## Alternate forms of the pythagorean identity

$1+{\mathrm{tan}}^{2}t={\mathrm{sec}}^{2}t$
${\mathrm{cot}}^{2}t+1={\mathrm{csc}}^{2}t$

## Using identities to relate trigonometric functions

If $\text{\hspace{0.17em}}\text{cos}\left(t\right)=\frac{12}{13}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ is in quadrant IV, as shown in [link] , find the values of the other five trigonometric functions.

We can find the sine using the Pythagorean Identity, $\text{\hspace{0.17em}}{\mathrm{cos}}^{2}t+{\mathrm{sin}}^{2}t=1,$ and the remaining functions by relating them to sine and cosine.

The sign of the sine depends on the y -values in the quadrant where the angle is located. Since the angle is in quadrant IV, where the y -values are negative, its sine is negative, $\text{\hspace{0.17em}}-\frac{5}{13}.$

The remaining functions can be calculated using identities relating them to sine and cosine.

$\begin{array}{l}\begin{array}{l}\begin{array}{l}\mathrm{tan}\text{\hspace{0.17em}}t=\frac{\mathrm{sin}\text{\hspace{0.17em}}t}{\mathrm{cos}\text{\hspace{0.17em}}t}=\frac{-\frac{5}{13}}{\frac{12}{13}}=-\frac{5}{12}\hfill \\ \begin{array}{l}\\ \mathrm{sec}\text{\hspace{0.17em}}t=\frac{1}{\mathrm{cos}\text{\hspace{0.17em}}t}=\frac{1}{\frac{12}{13}}=\frac{13}{12}\end{array}\hfill \end{array}\hfill \\ \mathrm{csc}\text{\hspace{0.17em}}t=\frac{1}{\mathrm{sin}\text{\hspace{0.17em}}t}=\frac{1}{-\frac{5}{13}}=-\frac{13}{5}\hfill \end{array}\\ \mathrm{cot}\text{\hspace{0.17em}}t=\frac{1}{\mathrm{tan}\text{\hspace{0.17em}}t}=\frac{1}{-\frac{5}{12}}=-\frac{12}{5}\end{array}$

If $\text{\hspace{0.17em}}\mathrm{sec}\left(t\right)=-\frac{17}{8}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}0 find the values of the other five functions.

$\mathrm{cos}\phantom{\rule{0.2em}{0ex}}t=-\frac{8}{17},\mathrm{sin}\phantom{\rule{0.2em}{0ex}}t=\frac{15}{17},\mathrm{tan}\phantom{\rule{0.2em}{0ex}}t=-\frac{15}{8}$
$\mathrm{csc}\phantom{\rule{0.2em}{0ex}}t=\frac{17}{15},\mathrm{cot}\phantom{\rule{0.2em}{0ex}}t=-\frac{8}{15}$

As we discussed in the chapter opening, a function that repeats its values in regular intervals is known as a periodic function . The trigonometric functions are periodic. For the four trigonometric functions, sine, cosine, cosecant and secant, a revolution of one circle, or $\text{\hspace{0.17em}}2\pi ,$ will result in the same outputs for these functions. And for tangent and cotangent, only a half a revolution will result in the same outputs.

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
"7"has an open circle and "10"has a filled in circle who can I have a set builder notation
Where do the rays point?
Spiro
x=-b+_Гb2-(4ac) ______________ 2a
I've run into this: x = r*cos(angle1 + angle2) Which expands to: x = r(cos(angle1)*cos(angle2) - sin(angle1)*sin(angle2)) The r value confuses me here, because distributing it makes: (r*cos(angle2))(cos(angle1) - (r*sin(angle2))(sin(angle1)) How does this make sense? Why does the r distribute once
so good
abdikarin
this is an identity when 2 adding two angles within a cosine. it's called the cosine sum formula. there is also a different formula when cosine has an angle minus another angle it's called the sum and difference formulas and they are under any list of trig identities
strategies to form the general term
carlmark
consider r(a+b) = ra + rb. The a and b are the trig identity.
Mike
How can you tell what type of parent function a graph is ?
generally by how the graph looks and understanding what the base parent functions look like and perform on a graph
William
if you have a graphed line, you can have an idea by how the directions of the line turns, i.e. negative, positive, zero
William
y=x will obviously be a straight line with a zero slope
William
y=x^2 will have a parabolic line opening to positive infinity on both sides of the y axis vice versa with y=-x^2 you'll have both ends of the parabolic line pointing downward heading to negative infinity on both sides of the y axis
William
y=x will be a straight line, but it will have a slope of one. Remember, if y=1 then x=1, so for every unit you rise you move over positively one unit. To get a straight line with a slope of 0, set y=1 or any integer.
Aaron
yes, correction on my end, I meant slope of 1 instead of slope of 0
William
what is f(x)=
I don't understand
Joe
Typically a function 'f' will take 'x' as input, and produce 'y' as output. As 'f(x)=y'. According to Google, "The range of a function is the complete set of all possible resulting values of the dependent variable (y, usually), after we have substituted the domain."
Thomas
Sorry, I don't know where the "Â"s came from. They shouldn't be there. Just ignore them. :-)
Thomas
Darius
Thanks.
Thomas
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Thomas
It is the Â that should not be there. It doesn't seem to show if encloses in quotation marks. "Â" or 'Â' ... Â
Thomas
Now it shows, go figure?
Thomas By By By By OpenStax By OpenStax By Saylor Foundation By OpenStax By Madison Christian By OpenStax By Rhodes By Richley Crapo By OpenStax