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5.2 Unit circle: sine and cosine functions  (Page 5/12)

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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

for the "hiking" mix, there are 1,000 pieces in the mix, containing 390.8 g of fat, and 165 g of protein. if there is the same amount of almonds as cashews, how many of each item is in the trail mix?
ADNAN Reply
linear speed of an object
Melissa Reply
an object is traveling around a circle with a radius of 13 meters .if in 20 seconds a central angle of 1/7 Radian is swept out what are the linear and angular speed of the object
Melissa
test
Matrix
how to find domain
Mohamed Reply
like this: (2)/(2-x) the aim is to see what will not be compatible with this rational expression. If x= 0 then the fraction is undefined since we cannot divide by zero. Therefore, the domain consist of all real numbers except 2.
Dan
define the term of domain
Moha
if a>0 then the graph is concave
Angel Reply
if a<0 then the graph is concave blank
Angel
what's a domain
Kamogelo Reply
The set of all values you can use as input into a function su h that the output each time will be defined, meaningful and real.
Spiro
how fast can i understand functions without much difficulty
Joe Reply
what is inequalities
Nathaniel
functions can be understood without a lot of difficulty. Observe the following: f(2) 2x - x 2(2)-2= 2 now observe this: (2,f(2)) ( 2, -2) 2(-x)+2 = -2 -4+2=-2
Dan
what is set?
Kelvin Reply
a colony of bacteria is growing exponentially doubling in size every 100 minutes. how much minutes will it take for the colony of bacteria to triple in size
Divya Reply
I got 300 minutes. is it right?
Patience
no. should be about 150 minutes.
Jason
It should be 158.5 minutes.
Mr
ok, thanks
Patience
100•3=300 300=50•2^x 6=2^x x=log_2(6) =2.5849625 so, 300=50•2^2.5849625 and, so, the # of bacteria will double every (100•2.5849625) = 258.49625 minutes
Thomas
158.5 This number can be developed by using algebra and logarithms. Begin by moving log(2) to the right hand side of the equation like this: t/100 log(2)= log(3) step 1: divide each side by log(2) t/100=1.58496250072 step 2: multiply each side by 100 to isolate t. t=158.49
Dan
what is the importance knowing the graph of circular functions?
Arabella Reply
can get some help basic precalculus
ismail Reply
What do you need help with?
Andrew
how to convert general to standard form with not perfect trinomial
Camalia Reply
can get some help inverse function
ismail
Rectangle coordinate
Asma Reply
how to find for x
Jhon Reply
it depends on the equation
Robert
yeah, it does. why do we attempt to gain all of them one side or the other?
Melissa
how to find x: 12x = 144 notice how 12 is being multiplied by x. Therefore division is needed to isolate x and whatever we do to one side of the equation we must do to the other. That develops this: x= 144/12 divide 144 by 12 to get x. addition: 12+x= 14 subtract 12 by each side. x =2
Dan
whats a domain
mike Reply
The domain of a function is the set of all input on which the function is defined. For example all real numbers are the Domain of any Polynomial function.
Spiro
Spiro; thanks for putting it out there like that, 😁
Melissa
foci (–7,–17) and (–7,17), the absolute value of the differenceof the distances of any point from the foci is 24.
Churlene Reply
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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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