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Sine and cosine functions

If t is a real number and a point ( x , y ) on the unit circle corresponds to an angle of t , then

cos t = x
sin t = y

Given a point P ( x , y ) on the unit circle corresponding to an angle of t , find the sine and cosine.

  1. The sine of t is equal to the y -coordinate of point P : sin t = y .
  2. The cosine of t is equal to the x -coordinate of point P :   cos t = x .

Finding function values for sine and cosine

Point P is a point on the unit circle corresponding to an angle of t , as shown in [link] . Find cos ( t ) and sin ( t ) .

Graph of a circle with angle t, radius of 1, and a terminal side that intersects the circle at the point (1/2, square root of 3 over 2).

We know that cos t is the x -coordinate of the corresponding point on the unit circle and sin t is the y -coordinate of the corresponding point on the unit circle. So:

x = cos t = 1 2 y = sin t = 3 2
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A certain angle t corresponds to a point on the unit circle at ( 2 2 , 2 2 ) as shown in [link] . Find cos t and sin t .

Graph of a circle with angle t, radius of 1, and a terminal side that intersects the circle at the point (negative square root of 2 over 2, square root of 2 over 2).

cos ( t ) = 2 2 , sin ( t ) = 2 2

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Finding sines and cosines of angles on an axis

For quadrantral angles, the corresponding point on the unit circle falls on the x- or y -axis. In that case, we can easily calculate cosine and sine from the values of x and y .

Calculating sines and cosines along an axis

Find cos ( 90° ) and sin ( 90° ) .

Moving 90° counterclockwise around the unit circle from the positive x -axis brings us to the top of the circle, where the ( x , y ) coordinates are (0, 1), as shown in [link] .

Graph of a circle with angle t, radius of 1, and a terminal side that intersects the circle at the point (0,1).

Using our definitions of cosine and sine,

x = cos t = cos ( 90° ) = 0 y = sin t = sin ( 90° ) = 1

The cosine of 90° is 0; the sine of 90° is 1.

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Find cosine and sine of the angle π .

cos ( π ) = 1 , sin ( π ) = 0

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The pythagorean identity

Now that we can define sine and cosine, we will learn how they relate to each other and the unit circle. Recall that the equation for the unit circle is x 2 + y 2 = 1. Because x = cos t and y = sin t , we can substitute for x and y to get cos 2 t + sin 2 t = 1. This equation, cos 2 t + sin 2 t = 1 , is known as the Pythagorean Identity . See [link] .

Graph of an angle t, with a point (x,y) on the unit circle. And equation showing the equivalence of 1, x^2 + y^2, and cos^2 t + sin^2 t.

We can use the Pythagorean Identity to find the cosine of an angle if we know the sine, or vice versa. However, because the equation yields two solutions, we need additional knowledge of the angle to choose the solution with the correct sign. If we know the quadrant where the angle is, we can easily choose the correct solution.

Pythagorean identity

The Pythagorean Identity    states that, for any real number t ,

cos 2 t + sin 2 t = 1

Given the sine of some angle t and its quadrant location, find the cosine of t .

  1. Substitute the known value of sin ( t ) into the Pythagorean Identity.
  2. Solve for cos ( t ) .
  3. Choose the solution with the appropriate sign for the x -values in the quadrant where t is located.

Finding a cosine from a sine or a sine from a cosine

If sin ( t ) = 3 7 and t is in the second quadrant, find cos ( t ) .

If we drop a vertical line from the point on the unit circle corresponding to t , we create a right triangle, from which we can see that the Pythagorean Identity is simply one case of the Pythagorean Theorem. See [link] .

Graph of a unit circle with an angle that intersects the circle at a point with the y-coordinate equal to 3/7.

Substituting the known value for sine into the Pythagorean Identity,

cos 2 ( t ) + sin 2 ( t ) = 1 cos 2 ( t ) + 9 49 = 1 cos 2 ( t ) = 40 49 cos ( t ) = ± 40 49 = ± 40 7 = ± 2 10 7

Because the angle is in the second quadrant, we know the x- value is a negative real number, so the cosine is also negative. So cos ( t ) = 2 10 7

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Questions & Answers

x=-b+_Гb2-(4ac) ______________ 2a
Ahlicia Reply
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
Carlos Reply
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
Brad
How can you tell what type of parent function a graph is ?
Mary Reply
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)=
Karim Reply
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
GREAT ANSWER THOUGH!!!
Darius
Thanks.
Thomas
Â
Thomas
It is the  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?
unknown Reply
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
Ef Reply
So a horizontal compression by factor of 1/2 is the same as a horizontal stretch by a factor of 2, right?
KARMEL Reply
The center is at (3,4) a focus is at (3,-1), and the lenght of the major axis is 26
Rima Reply
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
Can you please help me. Tomorrow is the deadline of my assignment then I don't know how to solve that
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.
Brittany Reply
how do you find the period of a sine graph
Imani Reply
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
Jhon Reply
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
Baptiste Reply
the sum of any two linear polynomial is what
Esther Reply
divide simplify each answer 3/2÷5/4
Momo Reply
divide simplify each answer 25/3÷5/12
Momo
how can are find the domain and range of a relations
austin Reply
the range is twice of the natural number which is the domain
Morolake
Practice Key Terms 4

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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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