# 7.3 Unit circle  (Page 5/11)

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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 $\text{\hspace{0.17em}}\alpha \text{\hspace{0.17em}}$ has the same sine value as angle $\text{\hspace{0.17em}}t;$ the cosine values are opposites. Angle $\text{\hspace{0.17em}}\beta \text{\hspace{0.17em}}$ has the same cosine value as angle $\text{\hspace{0.17em}}t;$ the sine values are opposites.

$\begin{array}{ccc}\mathrm{sin}\left(t\right)=\mathrm{sin}\left(\alpha \right)\hfill & \phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}& \mathrm{cos}\left(t\right)=-\mathrm{cos}\left(\alpha \right)\hfill \\ \mathrm{sin}\left(t\right)=-\mathrm{sin}\left(\beta \right)\hfill & \phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}& \mathrm{cos}\left(t\right)=\mathrm{cos}\left(\beta \right)\hfill \end{array}$

Recall that an angle’s reference angle is the acute angle, $\text{\hspace{0.17em}}t,$ formed by the terminal side of the angle $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ and the horizontal axis. A reference angle is always an angle between $\text{\hspace{0.17em}}0\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}90°,$ or $\text{\hspace{0.17em}}0\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\frac{\pi }{2}\text{\hspace{0.17em}}$ radians. As we can see from [link] , for any angle in quadrants II, III, or IV, there is a reference angle in quadrant I.

Given an angle between $\text{\hspace{0.17em}}0\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}2\pi ,$ 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 $\text{\hspace{0.17em}}|\pi -t|\text{\hspace{0.17em}}$ or $\text{\hspace{0.17em}}|180°-t|.$
3. For an angle in the fourth quadrant, the reference angle is $\text{\hspace{0.17em}}2\pi -t\text{\hspace{0.17em}}$ or $\text{\hspace{0.17em}}360°-t.$
4. If an angle is less than $\text{\hspace{0.17em}}0\text{\hspace{0.17em}}$ or greater than $\text{\hspace{0.17em}}2\pi ,$ add or subtract $\text{\hspace{0.17em}}2\pi \text{\hspace{0.17em}}$ as many times as needed to find an equivalent angle between $\text{\hspace{0.17em}}0\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}2\pi .$

## Finding a reference angle

Find the reference angle of $\text{\hspace{0.17em}}225°\text{\hspace{0.17em}}$ as shown in [link] .

Because $\text{\hspace{0.17em}}225°\text{\hspace{0.17em}}$ is in the third quadrant, the reference angle is

$|\left(180°-225°\right)|=|-45°|=45°$

Find the reference angle of $\text{\hspace{0.17em}}\frac{5\pi }{3}.$

$\frac{\pi }{3}$

## 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 $\text{\hspace{0.17em}}\left(x,y\right)\text{\hspace{0.17em}}$ 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 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.

f(x)=x/x+2 given g(x)=1+2x/1-x show that gf(x)=1+2x/3
proof
AUSTINE
sebd me some questions about anything ill solve for yall
how to solve x²=2x+8 factorization?
x=2x+8 x-2x=2x+8-2x x-2x=8 -x=8 -x/-1=8/-1 x=-8 prove: if x=-8 -8=2(-8)+8 -8=-16+8 -8=-8 (PROVEN)
Manifoldee
x=2x+8
Manifoldee
×=2x-8 minus both sides by 2x
Manifoldee
so, x-2x=2x+8-2x
Manifoldee
then cancel out 2x and -2x, cuz 2x-2x is obviously zero
Manifoldee
so it would be like this: x-2x=8
Manifoldee
then we all know that beside the variable is a number (1): (1)x-2x=8
Manifoldee
so we will going to minus that 1-2=-1
Manifoldee
so it would be -x=8
Manifoldee
so next step is to cancel out negative number beside x so we get positive x
Manifoldee
so by doing it you need to divide both side by -1 so it would be like this: (-1x/-1)=(8/-1)
Manifoldee
so -1/-1=1
Manifoldee
so x=-8
Manifoldee
Manifoldee
so we should prove it
Manifoldee
x=2x+8 x-2x=8 -x=8 x=-8 by mantu from India
mantu
lol i just saw its x²
Manifoldee
x²=2x-8 x²-2x=8 -x²=8 x²=-8 square root(x²)=square root(-8) x=sq. root(-8)
Manifoldee
I mean x²=2x+8 by factorization method
Kristof
I think x=-2 or x=4
Kristof
x= 2x+8 ×=8-2x - 2x + x = 8 - x = 8 both sides divided - 1 -×/-1 = 8/-1 × = - 8 //// from somalia
Mohamed
hii
Amit
how are you
Dorbor
well
Biswajit
can u tell me concepts
Gaurav
Find the possible value of 8.5 using moivre's theorem
which of these functions is not uniformly cintinuous on (0, 1)? sinx
which of these functions is not uniformly continuous on 0,1
solve this equation by completing the square 3x-4x-7=0
X=7
Muustapha
=7
mantu
x=7
mantu
3x-4x-7=0 -x=7 x=-7
Kr
x=-7
mantu
9x-16x-49=0 -7x=49 -x=7 x=7
mantu
what's the formula
Modress
-x=7
Modress
new member
siame
what is trigonometry
deals with circles, angles, and triangles. Usually in the form of Soh cah toa or sine, cosine, and tangent
Thomas
solve for me this equational y=2-x
what are you solving for
Alex
solve x
Rubben
you would move everything to the other side leaving x by itself. subtract 2 and divide -1.
Nikki
then I got x=-2
Rubben
it will b -y+2=x
Alex
goodness. I'm sorry. I will let Alex take the wheel.
Nikki
ouky thanks braa
Rubben
I think he drive me safe
Rubben
how to get 8 trigonometric function of tanA=0.5, given SinA=5/13? Can you help me?m
More example of algebra and trigo
What is Indices
If one side only of a triangle is given is it possible to solve for the unkown two sides?
cool
Rubben
kya
Khushnama
please I need help in maths
Okey tell me, what's your problem is?
Navin