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- Precalculus
- Trigonometric functions
- Unit circle: sine and cosine
Key equations
| Cosine |
|
| Sine |
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| Pythagorean Identity |
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Key concepts
- Finding the function values for the sine and cosine begins with drawing a unit circle, which is centered at the origin and has a radius of 1 unit.
- Using the unit circle, the sine of an angle
equals the
y -value of the endpoint on the unit circle of an arc of length
whereas the cosine of an angle
equals the
x -value of the endpoint. See
[link] .
- The sine and cosine values are most directly determined when the corresponding point on the unit circle falls on an axis. See
[link] .
- When the sine or cosine is known, we can use the Pythagorean Identity to find the other. The Pythagorean Identity is also useful for determining the sines and cosines of special angles. See
[link] .
- Calculators and graphing software are helpful for finding sines and cosines if the proper procedure for entering information is known. See
[link] .
- The domain of the sine and cosine functions is all real numbers.
- The range of both the sine and cosine functions is
- The sine and cosine of an angle have the same absolute value as the sine and cosine of its reference angle.
- The signs of the sine and cosine are determined from the
x - and
y -values in the quadrant of the original angle.
- An angle’s reference angle is the size angle,
formed by the terminal side of the angle
and the horizontal axis. See
[link] .
- Reference angles can be used to find the sine and cosine of the original angle. See
[link] .
- Reference angles can also be used to find the coordinates of a point on a circle. See
[link] .
Section exercises
Verbal
Discuss the difference between a coterminal angle and a reference angle.
Coterminal angles are angles that share the same terminal side. A reference angle is the size of the smallest acute angle,
formed by the terminal side of the angle
and the horizontal axis.
Got questions? Get instant answers now!
Algebraic
For the following exercises, use the given sign of the sine and cosine functions to find the quadrant in which the terminal point determined by
lies.
For the following exercises, find the exact value of each trigonometric function.
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?
linear speed of an object
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
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
if a<0 then the graph is concave blank
Angel
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
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
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
I got 300 minutes. is it right?
Patience
no. should be about 150 minutes.
Jason
It should be 158.5 minutes.
Mr
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?
can get some help basic precalculus
What do you need help with?
Andrew
how to convert general to standard form with not perfect trinomial
can get some help inverse function
ismail
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
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.
Source:
OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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