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Key equations

Cosine cos t = x
Sine sin t = y
Pythagorean Identity cos 2 t + sin 2 t = 1

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 t equals the y -value of the endpoint on the unit circle of an arc of length t whereas the cosine of an angle t 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 [ −1 , 1 ] .
  • 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, t , formed by the terminal side of the angle t 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

Describe the unit circle.

The unit circle is a circle of radius 1 centered at the origin.

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What do the x- and y- coordinates of the points on the unit circle represent?

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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, t , formed by the terminal side of the angle t and the horizontal axis.

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Explain how the cosine of an angle in the second quadrant differs from the cosine of its reference angle in the unit circle.

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Explain how the sine of an angle in the second quadrant differs from the sine of its reference angle in the unit circle.

The sine values are equal.

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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 t lies.

sin ( t ) < 0 and cos ( t ) < 0

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sin ( t ) > 0 and cos ( t ) > 0

I

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sin ( t ) > 0 and cos ( t ) < 0

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sin ( t ) > 0 and cos ( t ) > 0

IV

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For the following exercises, find the exact value of each trigonometric function.

Questions & Answers

A laser rangefinder is locked on a comet approaching Earth. The distance g(x), in kilometers, of the comet after x days, for x in the interval 0 to 30 days, is given by g(x)=250,000csc(π30x). Graph g(x) on the interval [0, 35]. Evaluate g(5)  and interpret the information. What is the minimum distance between the comet and Earth? When does this occur? To which constant in the equation does this correspond? Find and discuss the meaning of any vertical asymptotes.
Kaitlyn Reply
The sequence is {1,-1,1-1.....} has
amit Reply
circular region of radious
Kainat Reply
how can we solve this problem
Joel Reply
Sin(A+B) = sinBcosA+cosBsinA
Eseka Reply
Prove it
Eseka
Please prove it
Eseka
hi
Joel
June needs 45 gallons of punch. 2 different coolers. Bigger cooler is 5 times as large as smaller cooler. How many gallons in each cooler?
Arleathia Reply
7.5 and 37.5
Nando
find the sum of 28th term of the AP 3+10+17+---------
Prince Reply
I think you should say "28 terms" instead of "28th term"
Vedant
the 28th term is 175
Nando
192
Kenneth
if sequence sn is a such that sn>0 for all n and lim sn=0than prove that lim (s1 s2............ sn) ke hole power n =n
SANDESH Reply
write down the polynomial function with root 1/3,2,-3 with solution
Gift Reply
if A and B are subspaces of V prove that (A+B)/B=A/(A-B)
Pream Reply
write down the value of each of the following in surd form a)cos(-65°) b)sin(-180°)c)tan(225°)d)tan(135°)
Oroke Reply
Prove that (sinA/1-cosA - 1-cosA/sinA) (cosA/1-sinA - 1-sinA/cosA) = 4
kiruba Reply
what is the answer to dividing negative index
Morosi Reply
In a triangle ABC prove that. (b+c)cosA+(c+a)cosB+(a+b)cisC=a+b+c.
Shivam Reply
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Aaron Reply
Practice Key Terms 3

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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