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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

the gradient function of a curve is 2x+4 and the curve passes through point (1,4) find the equation of the curve
Kc Reply
1+cos²A/cos²A=2cosec²A-1
Ramesh Reply
test for convergence the series 1+x/2+2!/9x3
success Reply
a man walks up 200 meters along a straight road whose inclination is 30 degree.How high above the starting level is he?
Lhorren Reply
100 meters
Kuldeep
Find that number sum and product of all the divisors of 360
jancy Reply
answer
Ajith
exponential series
Naveen
what is subgroup
Purshotam Reply
Prove that: (2cos&+1)(2cos&-1)(2cos2&-1)=2cos4&+1
Macmillan Reply
e power cos hyperbolic (x+iy)
Vinay Reply
10y
Michael
tan hyperbolic inverse (x+iy)=alpha +i bita
Payal Reply
prove that cos(π/6-a)*cos(π/3+b)-sin(π/6-a)*sin(π/3+b)=sin(a-b)
Tejas Reply
why {2kπ} union {kπ}={kπ}?
Huy Reply
why is {2kπ} union {kπ}={kπ}? when k belong to integer
Huy
if 9 sin theta + 40 cos theta = 41,prove that:41 cos theta = 41
Trilochan Reply
what is complex numbers
Ayushi Reply
Please you teach
Dua
Yes
ahmed
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Dua
give me treganamentry question
Anshuman Reply
Solve 2cos x + 3sin x = 0.5
shobana 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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