# 5.2 Unit circle: sine and cosine functions

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In this section, you will:
• Find function values for the sine and cosine of and $\text{\hspace{0.17em}}{60°}^{}\text{or}\left(\frac{\pi }{3}\right).$
• Identify the domain and range of sine and cosine functions.
• Use reference angles to evaluate trigonometric functions.

Looking for a thrill? Then consider a ride on the Singapore Flyer, the world’s tallest Ferris wheel. Located in Singapore, the Ferris wheel soars to a height of 541 feet—a little more than a tenth of a mile! Described as an observation wheel, riders enjoy spectacular views as they travel from the ground to the peak and down again in a repeating pattern. In this section, we will examine this type of revolving motion around a circle. To do so, we need to define the type of circle first, and then place that circle on a coordinate system. Then we can discuss circular motion in terms of the coordinate pairs.

## Finding function values for the sine and cosine

To define our trigonometric functions, we begin by drawing a unit circle, a circle centered at the origin with radius 1, as shown in [link] . The angle (in radians) that $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ intercepts forms an arc of length $\text{\hspace{0.17em}}s.\text{\hspace{0.17em}}$ Using the formula $\text{\hspace{0.17em}}s=rt,$ and knowing that $r=1,$ we see that for a unit circle    , $\text{\hspace{0.17em}}s=t.\text{\hspace{0.17em}}$

Recall that the x- and y- axes divide the coordinate plane into four quarters called quadrants. We label these quadrants to mimic the direction a positive angle would sweep. The four quadrants are labeled I, II, III, and IV.

For any angle $\text{\hspace{0.17em}}t,$ we can label the intersection of the terminal side and the unit circle as by its coordinates, $\text{\hspace{0.17em}}\left(x,y\right).\text{\hspace{0.17em}}$ The coordinates $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ will be the outputs of the trigonometric functions $\text{\hspace{0.17em}}f\left(t\right)=\mathrm{cos}\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}f\left(t\right)=\mathrm{sin}\text{\hspace{0.17em}}t,$ respectively. This means $\text{\hspace{0.17em}}x=\mathrm{cos}\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}y=\mathrm{sin}\text{\hspace{0.17em}}t.\text{\hspace{0.17em}}$

## Unit circle

A unit circle    has a center at $\text{\hspace{0.17em}}\left(0,0\right)\text{\hspace{0.17em}}$ and radius $\text{\hspace{0.17em}}1\text{\hspace{0.17em}}$ . In a unit circle, the length of the intercepted arc is equal to the radian measure of the central angle $\text{\hspace{0.17em}}1.$

Let $\text{\hspace{0.17em}}\left(x,y\right)\text{\hspace{0.17em}}$ be the endpoint on the unit circle of an arc of arc length $\text{\hspace{0.17em}}s.\text{\hspace{0.17em}}$ The $\text{\hspace{0.17em}}\left(x,y\right)\text{\hspace{0.17em}}$ coordinates of this point can be described as functions of the angle.

## Defining sine and cosine functions

Now that we have our unit circle labeled, we can learn how the $\text{\hspace{0.17em}}\left(x,y\right)\text{\hspace{0.17em}}$ coordinates relate to the arc length    and angle    . The sine function    relates a real number $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ to the y -coordinate of the point where the corresponding angle intercepts the unit circle. More precisely, the sine of an angle $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ equals the y -value of the endpoint on the unit circle of an arc of length $\text{\hspace{0.17em}}t.\text{\hspace{0.17em}}$ In [link] , the sine is equal to $\text{\hspace{0.17em}}y.\text{\hspace{0.17em}}$ Like all functions, the sine function has an input and an output. Its input is the measure of the angle; its output is the y -coordinate of the corresponding point on the unit circle.

The cosine function    of an angle $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ equals the x -value of the endpoint on the unit circle of an arc of length $\text{\hspace{0.17em}}t.\text{\hspace{0.17em}}$ In [link] , the cosine is equal to $\text{\hspace{0.17em}}x.\text{\hspace{0.17em}}$

Because it is understood that sine and cosine are functions, we do not always need to write them with parentheses: $\mathrm{sin}\text{\hspace{0.17em}}t$ is the same as $\mathrm{sin}\left(t\right)$ and $\mathrm{cos}\text{\hspace{0.17em}}t$ is the same as $\mathrm{cos}\left(t\right).$ Likewise, $\text{\hspace{0.17em}}{\mathrm{cos}}^{2}t\text{\hspace{0.17em}}$ is a commonly used shorthand notation for ${\left(\mathrm{cos}\left(t\right)\right)}^{2}.\text{\hspace{0.17em}}$ Be aware that many calculators and computers do not recognize the shorthand notation. When in doubt, use the extra parentheses when entering calculations into a calculator or computer.

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
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
Â
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
what is this?
i do not understand anything
unknown
lol...it gets better
Darius
I've been struggling so much through all of this. my final is in four weeks 😭
Tiffany
this book is an excellent resource! have you guys ever looked at the online tutoring? there's one that is called "That Tutor Guy" and he goes over a lot of the concepts
Darius
thank you I have heard of him. I should check him out.
Tiffany
is there any question in particular?
Joe
I have always struggled with math. I get lost really easy, if you have any advice for that, it would help tremendously.
Tiffany
Sure, are you in high school or college?
Darius
Hi, apologies for the delayed response. I'm in college.
Tiffany
how to solve polynomial using a calculator
So a horizontal compression by factor of 1/2 is the same as a horizontal stretch by a factor of 2, right?
The center is at (3,4) a focus is at (3,-1), and the lenght of the major axis is 26
The center is at (3,4) a focus is at (3,-1) and the lenght of the major axis is 26 what will be the answer?
Rima
I done know
Joe
What kind of answer is that😑?
Rima
I had just woken up when i got this message
Joe
Rima
i have a question.
Abdul
how do you find the real and complex roots of a polynomial?
Abdul
@abdul with delta maybe which is b(square)-4ac=result then the 1st root -b-radical delta over 2a and the 2nd root -b+radical delta over 2a. I am not sure if this was your question but check it up
Nare
This is the actual question: Find all roots(real and complex) of the polynomial f(x)=6x^3 + x^2 - 4x + 1
Abdul
@Nare please let me know if you can solve it.
Abdul
I have a question
juweeriya
hello guys I'm new here? will you happy with me
mustapha
The average annual population increase of a pack of wolves is 25.
how do you find the period of a sine graph
Period =2π if there is a coefficient (b), just divide the coefficient by 2π to get the new period
Am
if not then how would I find it from a graph
Imani
by looking at the graph, find the distance between two consecutive maximum points (the highest points of the wave). so if the top of one wave is at point A (1,2) and the next top of the wave is at point B (6,2), then the period is 5, the difference of the x-coordinates.
Am
you could also do it with two consecutive minimum points or x-intercepts
Am
I will try that thank u
Imani
Case of Equilateral Hyperbola
ok
Zander
ok
Shella
f(x)=4x+2, find f(3)
Benetta
f(3)=4(3)+2 f(3)=14
lamoussa
14
Vedant
pre calc teacher: "Plug in Plug in...smell's good" f(x)=14
Devante
8x=40
Chris
Explain why log a x is not defined for a < 0
the sum of any two linear polynomial is what
Momo
how can are find the domain and range of a relations
the range is twice of the natural number which is the domain
Morolake