# 5.2 Unit circle: sine and cosine functions  (Page 4/12)

 Page 4 / 12

From the Pythagorean Theorem, we get

${x}^{2}+{y}^{2}=1$

Substituting $\text{\hspace{0.17em}}x=\frac{1}{2},$ we get

${\left(\frac{1}{2}\right)}^{2}+{y}^{2}=1$

Solving for $\text{\hspace{0.17em}}y,$ we get

Since $\text{\hspace{0.17em}}t=\frac{\pi }{3}\text{\hspace{0.17em}}$ has the terminal side in quadrant I where the y- coordinate is positive, we choose $\text{\hspace{0.17em}}y=\frac{\sqrt{3}}{2},$ the positive value.

At $\text{\hspace{0.17em}}t=\frac{\pi }{3}\text{\hspace{0.17em}}$ (60°), the $\text{\hspace{0.17em}}\left(x,y\right)\text{\hspace{0.17em}}$ coordinates for the point on a circle of radius $\text{\hspace{0.17em}}1\text{\hspace{0.17em}}$ at an angle of $\text{\hspace{0.17em}}60°\text{\hspace{0.17em}}$ are $\text{\hspace{0.17em}}\left(\frac{1}{2},\frac{\sqrt{3}}{2}\right),$ so we can find the sine and cosine.

$\begin{array}{l}\left(x,y\right)=\left(\frac{1}{2},\frac{\sqrt{3}}{2}\right)\hfill \\ x=\frac{1}{2},y=\frac{\sqrt{3}}{2}\hfill \\ \mathrm{cos}\text{\hspace{0.17em}}t=\frac{1}{2},\mathrm{sin}\text{\hspace{0.17em}}t=\frac{\sqrt{3}}{2}\hfill \end{array}$

We have now found the cosine and sine values for all of the most commonly encountered angles in the first quadrant of the unit circle. [link] summarizes these values.

 Angle 0 $\text{\hspace{0.17em}}\frac{\pi }{6},$ or 30 $\text{\hspace{0.17em}}\frac{\pi }{4},$ or 45° $\text{\hspace{0.17em}}\frac{\pi }{3},$ or 60° $\text{\hspace{0.17em}}\frac{\pi }{2},$ or 90° Cosine 1 $\frac{\sqrt{3}}{2}$ $\frac{\sqrt{2}}{2}\text{\hspace{0.17em}}$ $\frac{1}{2}$ 0 Sine 0 $\text{\hspace{0.17em}}\frac{1}{2}\text{\hspace{0.17em}}$ $\frac{\sqrt{2}}{2}\text{\hspace{0.17em}}$ $\frac{\sqrt{3}}{2}$ 1

[link] shows the common angles in the first quadrant of the unit circle.

## Using a calculator to find sine and cosine

To find the cosine and sine of angles other than the special angles , we turn to a computer or calculator. Be aware : Most calculators can be set into “degree” or “radian” mode, which tells the calculator the units for the input value. When we evaluate $\text{\hspace{0.17em}}\mathrm{cos}\left(30\right)\text{\hspace{0.17em}}$ on our calculator, it will evaluate it as the cosine of 30 degrees if the calculator is in degree mode, or the cosine of 30 radians if the calculator is in radian mode.

Given an angle in radians, use a graphing calculator to find the cosine.

1. If the calculator has degree mode and radian mode, set it to radian mode.
2. Press the COS key.
3. Enter the radian value of the angle and press the close-parentheses key ")".
4. Press ENTER.

## Using a graphing calculator to find sine and cosine

Evaluate $\text{\hspace{0.17em}}\mathrm{cos}\left(\frac{5\pi }{3}\right)\text{\hspace{0.17em}}$ using a graphing calculator or computer.

Enter the following keystrokes:

$\mathrm{cos}\left(\frac{5\pi }{3}\right)=0.5$

Evaluate $\text{\hspace{0.17em}}\mathrm{sin}\left(\frac{\pi }{3}\right).$

approximately 0.866025403

## Identifying the domain and range of sine and cosine functions

Now that we can find the sine and cosine of an angle, we need to discuss their domains and ranges. What are the domains of the sine and cosine functions? That is, what are the smallest and largest numbers that can be inputs of the functions? Because angles smaller than 0 and angles larger than $\text{\hspace{0.17em}}2\pi \text{\hspace{0.17em}}$ can still be graphed on the unit circle and have real values of $\text{\hspace{0.17em}}x,y,$ and $\text{\hspace{0.17em}}r,$ there is no lower or upper limit to the angles that can be inputs to the sine and cosine functions. The input to the sine and cosine functions is the rotation from the positive x -axis, and that may be any real number.

What are the ranges of the sine and cosine functions? What are the least and greatest possible values for their output? We can see the answers by examining the unit circle    , as shown in [link] . The bounds of the x -coordinate are $\text{\hspace{0.17em}}\left[-1,1\right].\text{\hspace{0.17em}}$ The bounds of the y -coordinate are also $\text{\hspace{0.17em}}\left[-1,1\right].\text{\hspace{0.17em}}$ Therefore, the range of both the sine and cosine functions is $\text{\hspace{0.17em}}\left[-1,1\right].\text{\hspace{0.17em}}$

## Finding reference angles

We have discussed finding the sine and cosine for angles in the first quadrant, but what if our angle is in another quadrant? For any given angle in the first quadrant, there is an angle in the second quadrant with the same sine value. Because the sine value is the y -coordinate on the unit circle, the other angle with the same sine will share the same y -value, but have the opposite x -value. Therefore, its cosine value will be the opposite of the first angle’s cosine value.

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