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Given that sin α = 4 5 and α lies in quadrant IV, find the exact value of cos ( α 2 ) .

2 5

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Finding the measurement of a half angle

Now, we will return to the problem posed at the beginning of the section. A bicycle ramp is constructed for high-level competition with an angle of θ formed by the ramp and the ground. Another ramp is to be constructed half as steep for novice competition. If tan θ = 5 3 for higher-level competition, what is the measurement of the angle for novice competition?

Since the angle for novice competition measures half the steepness of the angle for the high level competition, and tan θ = 5 3 for high competition, we can find cos θ from the right triangle and the Pythagorean theorem so that we can use the half-angle identities. See [link] .

3 2 + 5 2 = 34 c = 34
Image of a right triangle with sides 3, 5, and rad34. Rad 34 is the hypotenuse, and 3 is the base. The angle formed by the hypotenuse and base is theta. The angle between the side of length 3 and side of length 5 is a right angle.

We see that cos θ = 3 34 = 3 34 34 . We can use the half-angle formula for tangent: tan θ 2 = 1 cos θ 1 + cos θ . Since tan θ is in the first quadrant, so is tan θ 2 .

tan θ 2 = 1 3 34 34 1 + 3 34 34 = 34 3 34 34 34 + 3 34 34 = 34 3 34 34 + 3 34 0.57

We can take the inverse tangent to find the angle: tan 1 ( 0.57 ) 29.7° . So the angle of the ramp for novice competition is 29.7° .

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Access these online resources for additional instruction and practice with double-angle, half-angle, and reduction formulas.

Key equations

Double-angle formulas sin ( 2 θ ) = 2 sin θ cos θ cos ( 2 θ ) = cos 2 θ sin 2 θ = 1 2 sin 2 θ = 2 cos 2 θ 1 tan ( 2 θ ) = 2 tan θ 1 tan 2 θ
Reduction formulas sin 2 θ = 1 cos ( 2 θ ) 2 cos 2 θ = 1 + cos ( 2 θ ) 2 tan 2 θ = 1 cos ( 2 θ ) 1 + cos ( 2 θ )
Half-angle formulas sin α 2 = ± 1 cos α 2 cos α 2 = ± 1 + cos α 2 tan α 2 = ± 1 cos α 1 + cos α = sin α 1 + cos α = 1 cos α sin α

Key concepts

  • Double-angle identities are derived from the sum formulas of the fundamental trigonometric functions: sine, cosine, and tangent. See [link] , [link] , [link] , and [link] .
  • Reduction formulas are especially useful in calculus, as they allow us to reduce the power of the trigonometric term. See [link] and [link] .
  • Half-angle formulas allow us to find the value of trigonometric functions involving half-angles, whether the original angle is known or not. See [link] , [link] , and [link] .

Section exercises

Verbal

Explain how to determine the reduction identities from the double-angle identity cos ( 2 x ) = cos 2 x sin 2 x .

Use the Pythagorean identities and isolate the squared term.

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Explain how to determine the double-angle formula for tan ( 2 x ) using the double-angle formulas for cos ( 2 x ) and sin ( 2 x ) .

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We can determine the half-angle formula for tan ( x 2 ) = 1 cos x 1 + cos x by dividing the formula for sin ( x 2 ) by cos ( x 2 ) . Explain how to determine two formulas for tan ( x 2 ) that do not involve any square roots.

1 cos x sin x , sin x 1 + cos x , multiplying the top and bottom by 1 cos x and 1 + cos x , respectively.

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For the half-angle formula given in the previous exercise for tan ( x 2 ) , explain why dividing by 0 is not a concern. (Hint: examine the values of cos x necessary for the denominator to be 0.)

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Algebraic

For the following exercises, find the exact values of a) sin ( 2 x ) , b) cos ( 2 x ) , and c) tan ( 2 x ) without solving for x .

If sin x = 1 8 , and x is in quadrant I.

a) 3 7 32 b) 31 32 c) 3 7 31

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If cos x = 2 3 , and x is in quadrant I.

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If cos x = 1 2 , and x is in quadrant III.

a) 3 2 b) 1 2 c) 3

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Questions & Answers

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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.
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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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