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Likewise, there will be an angle in the fourth quadrant with the same cosine as the original angle. The angle with the same cosine will share the same x -value but will have the opposite y -value. Therefore, its sine value will be the opposite of the original angle’s sine value.

As shown in [link] , angle α has the same sine value as angle t ; the cosine values are opposites. Angle β has the same cosine value as angle t ; the sine values are opposites.

sin ( t ) = sin ( α ) and cos ( t ) = cos ( α ) sin ( t ) = sin ( β ) and cos ( t ) = cos ( β )
Graph of two side by side circles. First graph has circle with angle t and angle alpha with radius r. Second graph has circle with angle t and angle beta inscribed with radius r.

Recall that an angle’s reference angle    is the acute angle, t , formed by the terminal side of the angle t and the horizontal axis. A reference angle is always an angle between 0 and 90° , or 0 and π 2 radians. As we can see from [link] , for any angle in quadrants II, III, or IV, there is a reference angle in quadrant I.

Four side by side graphs. First graph shows an angle of t in quadrant 1 in it's normal position. Second graph shows an angle of t in quadrant 2 due to a rotation of pi minus t. Third graph shows an angle of t in quadrant 3 due to a rotation of t minus pi. Fourth graph shows an angle of t in quadrant 4 due to a rotation of two pi minus t.

Given an angle between 0 and 2 π , find its reference angle.

  1. An angle in the first quadrant is its own reference angle.
  2. For an angle in the second or third quadrant, the reference angle is | π t | or | 180° −t | .
  3. For an angle in the fourth quadrant, the reference angle is 2 π t or 360° −t .
  4. If an angle is less than 0 or greater than 2 π , add or subtract 2 π as many times as needed to find an equivalent angle between 0 and 2 π .

Finding a reference angle

Find the reference angle of 225° as shown in [link] .

Graph of circle with 225 degree angle inscribed.

Because 225° is in the third quadrant, the reference angle is

| ( 180° −225° ) | = | 45° | = 45°
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Find the reference angle of 5 π 3 .

π 3

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Using reference angles

Now let’s take a moment to reconsider the Ferris wheel introduced at the beginning of this section. Suppose a rider snaps a photograph while stopped twenty feet above ground level. The rider then rotates three-quarters of the way around the circle. What is the rider’s new elevation? To answer questions such as this one, we need to evaluate the sine or cosine functions at angles that are greater than 90 degrees or at a negative angle    . Reference angles make it possible to evaluate trigonometric functions for angles outside the first quadrant. They can also be used to find ( x , y ) coordinates for those angles. We will use the reference angle    of the angle of rotation combined with the quadrant in which the terminal side of the angle lies.

Using reference angles to evaluate trigonometric functions

We can find the cosine and sine of any angle in any quadrant if we know the cosine or sine of its reference angle. The absolute values of the cosine and sine of an angle are the same as those of the reference angle. The sign depends on the quadrant of the original angle. The cosine will be positive or negative depending on the sign of the x -values in that quadrant. The sine will be positive or negative depending on the sign of the y -values in that quadrant.

Using reference angles to find cosine and sine

Angles have cosines and sines with the same absolute value as cosines and sines of their reference angles. The sign (positive or negative) can be determined from the quadrant of the angle.

Given an angle in standard position, find the reference angle, and the cosine and sine of the original angle.

  1. Measure the angle between the terminal side of the given angle and the horizontal axis. That is the reference angle.
  2. Determine the values of the cosine and sine of the reference angle.
  3. Give the cosine the same sign as the x -values in the quadrant of the original angle.
  4. Give the sine the same sign as the y -values in the quadrant of the original angle.

Questions & Answers

difference between calculus and pre calculus?
Asma Reply
give me an example of a problem so that I can practice answering
Jenefa Reply
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?
CJ Reply
I want to learn about the law of exponent
Quera Reply
explain this
Hinderson Reply
what is functions?
Angel Reply
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
If the plane intersects the cone (either above or below) horizontally, what figure will be created?
Feemark Reply
can you not take the square root of a negative number
Sharon Reply
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
Elaine Reply
can I get some pretty basic questions
Ama Reply
In what way does set notation relate to function notation
Ama
is precalculus needed to take caculus
Amara Reply
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
Mars Reply
what is the domain of f(x)=x-4/x^2-2x-15 then
Conney Reply
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
jeric Reply
Don't think that you can.
Elliott
By using some imaginary no.
Tanmay
how do you provided cos⁴x-sin⁴x = cos²x-sin²x are equal
jeric Reply
What are the question marks for?
Elliott
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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