# 7.3 Unit circle  (Page 5/11)

 Page 5 / 11

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.

#### Questions & Answers

bsc F. y algebra and trigonometry pepper 2
given that x= 3/5 find sin 3x
4
DB
remove any signs and collect terms of -2(8a-3b-c)
-16a+6b+2c
Will
is that a real answer
Joeval
(x2-2x+8)-4(x2-3x+5)
sorry
Miranda
x²-2x+9-4x²+12x-20 -3x²+10x+11
Miranda
x²-2x+9-4x²+12x-20 -3x²+10x+11
Miranda
(X2-2X+8)-4(X2-3X+5)=0 ?
master
The anwser is imaginary number if you want to know The anwser of the expression you must arrange The expression and use quadratic formula To find the answer
master
The anwser is imaginary number if you want to know The anwser of the expression you must arrange The expression and use quadratic formula To find the answer
master
Y
master
X2-2X+8-4X2+12X-20=0 (X2-4X2)+(-2X+12X)+(-20+8)= 0 -3X2+10X-12=0 3X2-10X+12=0 Use quadratic formula To find the answer answer (5±Root11i)/3
master
Soo sorry (5±Root11* i)/3
master
x2-2x+8-4x2+12x-20 x2-4x2-2x+12x+8-20 -3x2+10x-12 now you can find the answer using quadratic
Mukhtar
explain and give four example of hyperbolic function
What is the correct rational algebraic expression of the given "a fraction whose denominator is 10 more than the numerator y?
y/y+10
Mr
Find nth derivative of eax sin (bx + c).
Find area common to the parabola y2 = 4ax and x2 = 4ay.
Anurag
A rectangular garden is 25ft wide. if its area is 1125ft, what is the length of the garden
to find the length I divide the area by the wide wich means 1125ft/25ft=45
Miranda
thanks
Jhovie
What do you call a relation where each element in the domain is related to only one value in the range by some rules?
A banana.
Yaona
given 4cot thither +3=0and 0°<thither <180° use a sketch to determine the value of the following a)cos thither
what are you up to?
nothing up todat yet
Miranda
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jai
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jai
Miranda Drice
jai
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jai
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Miranda
I am living in india
jai
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Miranda
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I think the formula for calculating algebraic is the statement of the equality of two expression stimulate by a set of addition, multiplication, soustraction, division, raising to a power and extraction of Root. U believe by having those in the equation you will be in measure to calculate it
Miranda
state and prove Cayley hamilton therom
hello
Propessor
hi
Miranda
the Cayley hamilton Theorem state if A is a square matrix and if f(x) is its characterics polynomial then f(x)=0 in another ways evey square matrix is a root of its chatacteristics polynomial.
Miranda
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jai
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jai
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Propessor
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jai
What is algebra
algebra is a branch of the mathematics to calculate expressions follow.
Miranda
Miranda Drice would you mind teaching me mathematics? I think you are really good at math. I'm not good at it. In fact I hate it. 😅😅😅
Jeffrey
lolll who told you I'm good at it
Miranda
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Jeffrey
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Miranda
but seriously, Im really bad at math. And I hate it. But you see, I downloaded this app two months ago hoping to master it.
Jeffrey
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Miranda
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Miranda
haha. already finished college
Jeffrey
how about you? what grade are you now?
Jeffrey
I'm going to 11grade
Miranda
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Miranda
gotta practice, holmie
Steve
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Steve
I don't know why. But Im trying to like it.
Jeffrey
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Jeffrey
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Miranda
what is the solution of the given equation?
which equation
Miranda
I dont know. lol
Jeffrey
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Miranda
Jeffrey