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Key equations

Tangent function tan t = sin t cos t
Secant function sec t = 1 cos t
Cosecant function csc t = 1 sin t
Cotangent function cot t = 1 tan t = cos t sin t

Key concepts

  • The tangent of an angle is the ratio of the y -value to the x -value of the corresponding point on the unit circle.
  • The secant, cotangent, and cosecant are all reciprocals of other functions. The secant is the reciprocal of the cosine function, the cotangent is the reciprocal of the tangent function, and the cosecant is the reciprocal of the sine function.
  • The six trigonometric functions can be found from a point on the unit circle. See [link] .
  • Trigonometric functions can also be found from an angle. See [link] .
  • Trigonometric functions of angles outside the first quadrant can be determined using reference angles. See [link] .
  • A function is said to be even if f ( x ) = f ( x ) and odd if f ( x ) = f ( x ) for all x in the domain of f.
  • Cosine and secant are even; sine, tangent, cosecant, and cotangent are odd.
  • Even and odd properties can be used to evaluate trigonometric functions. See [link] .
  • The Pythagorean Identity makes it possible to find a cosine from a sine or a sine from a cosine.
  • Identities can be used to evaluate trigonometric functions. See [link] and [link] .
  • Fundamental identities such as the Pythagorean Identity can be manipulated algebraically to produce new identities. See [link] .The trigonometric functions repeat at regular intervals.
  • The period P of a repeating function f is the smallest interval such that f ( x + P ) = f ( x ) for any value of x .
  • The values of trigonometric functions can be found by mathematical analysis. See [link] and [link] .
  • To evaluate trigonometric functions of other angles, we can use a calculator or computer software. See [link] .

Section exercises

Verbal

On an interval of [ 0 , 2 π ) , can the sine and cosine values of a radian measure ever be equal? If so, where?

Yes, when the reference angle is π 4 and the terminal side of the angle is in quadrants I and III. Thus, a x = π 4 , 5 π 4 , the sine and cosine values are equal.

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What would you estimate the cosine of π degrees to be? Explain your reasoning.

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For any angle in quadrant II, if you knew the sine of the angle, how could you determine the cosine of the angle?

Substitute the sine of the angle in for y in the Pythagorean Theorem x 2 + y 2 = 1. Solve for x and take the negative solution.

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Describe the secant function.

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Tangent and cotangent have a period of π . What does this tell us about the output of these functions?

The outputs of tangent and cotangent will repeat every π units.

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Algebraic

For the following exercises, find the exact value of each expression.

For the following exercises, use reference angles to evaluate the expression.

sec 7 π 6

2 3 3

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If sin t = 3 4 , and t is in quadrant II, find cos t , sec t , csc t , tan t , and cot t .

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If cos t = 1 3 , and t is in quadrant III, find sin t , sec t , csc t , tan t , and cot t .

sin t = 2 2 3 , sec t = 3 , csc t = 3 2 4 , tan t = 2 2 , cot t = 2 4

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Questions & Answers

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
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
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
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write down the polynomial function with root 1/3,2,-3 with solution
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if A and B are subspaces of V prove that (A+B)/B=A/(A-B)
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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
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Morosi Reply
In a triangle ABC prove that. (b+c)cosA+(c+a)cosB+(a+b)cisC=a+b+c.
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the polar co-ordinate of the point (-1, -1)
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Practice Key Terms 6

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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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