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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 . Thus,

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


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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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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Questions & Answers

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Hey I am new to precalculus, and wanted clarification please on what sine is as I am floored by the terms in this app? I don't mean to sound stupid but I have only completed up to college algebra.
rachel Reply
I don't know if you are looking for a deeper answer or not, but the sine of an angle in a right triangle is the length of the opposite side to the angle in question divided by the length of the hypotenuse of said triangle.
can you give me sir tips to quickly understand precalculus. Im new too in that topic. Thanks
if you remember sine, cosine, and tangent from geometry, all the relationships are the same but they use x y and r instead (x is adjacent, y is opposite, and r is hypotenuse).
it is better to use unit circle than triangle .triangle is only used for acute angles but you can begin with. Download any application named"unit circle" you find in it all you need. unit circle is a circle centred at origine (0;0) with radius r= 1.
What is domain
the standard equation of the ellipse that has vertices (0,-4)&(0,4) and foci (0, -15)&(0,15) it's standard equation is x^2 + y^2/16 =1 tell my why is it only x^2? why is there no a^2?
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This term is plural for a focus, it is used for conic sections. For more detail or other math questions. I recommend researching on "Khan academy" or watching "The Organic Chemistry Tutor" YouTube channel.
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Confunction Identity
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For each year t, the population of a forest of trees is represented by the function A(t) = 117(1.029)t. In a neighboring forest, the population of the same type of tree is represented by the function B(t) = 86(1.025)t.
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It's just like any other number. The important thing to know is that they exist and can be used in computations like any number.
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Good call Scott. Also radar signals I believe.
They are used in any profession where the phase of a waveform has to be accounted for in the calculations. Imagine two electrical signals in a wire that are out of phase by 90°. At some times they will interfere constructively, others destructively. Complex numbers simplify those equations
Practice Key Terms 3

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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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