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7.3 Double-angle, half-angle, and reduction formulas  (Page 9/8)

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In this section, you will:
  • Use double-angle formulas to find exact values.
  • Use double-angle formulas to verify identities.
  • Use reduction formulas to simplify an expression.
  • Use half-angle formulas to find exact values.
Picture of two bicycle ramps, one with a steep slope and one with a gentle slope.
Bicycle ramps for advanced riders have a steeper incline than those designed for novices.

Bicycle ramps made for competition (see [link] ) must vary in height depending on the skill level of the competitors. For advanced competitors, the angle formed by the ramp and the ground should be   θ   such that   tan   θ = 5 3 .   The angle is divided in half for novices. What is the steepness of the ramp for novices? In this section, we will investigate three additional categories of identities that we can use to answer questions such as this one.

Using double-angle formulas to find exact values

In the previous section, we used addition and subtraction formulas for trigonometric functions. Now, we take another look at those same formulas. The double-angle formulas    are a special case of the sum formulas, where   α = β .   Deriving the double-angle formula for sine begins with the sum formula,

sin ( α + β ) = sin   α   cos   β + cos   α   sin   β

If we let   α = β = θ , then we have

sin ( θ + θ ) = sin   θ   cos   θ + cos   θ   sin   θ      sin ( 2 θ ) = 2 sin   θ   cos   θ

Deriving the double-angle for cosine gives us three options. First, starting from the sum formula,   cos ( α + β ) = cos   α   cos   β − sin   α   sin   β , and letting   α = β = θ , we have

cos ( θ + θ ) = cos   θ   cos   θ − sin   θ sin   θ      cos ( 2 θ ) = cos 2 θ − sin 2 θ

Using the Pythagorean properties, we can expand this double-angle formula for cosine and get two more interpretations. The first one is:

cos ( 2 θ ) = cos 2 θ − sin 2 θ              = ( 1 − sin 2 θ ) − sin 2 θ              = 1 − 2 sin 2 θ

The second interpretation is:

cos ( 2 θ ) = cos 2 θ − sin 2 θ              = cos 2 θ − ( 1 − cos 2 θ )              = 2   cos 2 θ − 1

Similarly, to derive the double-angle formula for tangent, replacing   α = β = θ   in the sum formula gives

tan ( α + β ) = tan   α + tan   β 1 − tan   α   tan   β tan ( θ + θ ) = tan   θ + tan   θ 1 − tan   θ   tan   θ tan ( 2 θ ) = 2 tan   θ 1 − tan 2 θ

Double-angle formulas

The double-angle formulas    are summarized as follows:

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

Given the tangent of an angle and the quadrant in which it is located, use the double-angle formulas to find the exact value.

  1. Draw a triangle to reflect the given information.
  2. Determine the correct double-angle formula.
  3. Substitute values into the formula based on the triangle.
  4. Simplify.

Using a double-angle formula to find the exact value involving tangent

Given that   tan   θ = − 3 4   and   θ   is in quadrant II, find the following:

  1. sin ( 2 θ )
  2. cos ( 2 θ )
  3. tan ( 2 θ )

If we draw a triangle to reflect the information given, we can find the values needed to solve the problems on the image. We are given   tan   θ = − 3 4 , such that   θ   is in quadrant II. The tangent of an angle is equal to the opposite side over the adjacent side, and because   θ   is in the second quadrant, the adjacent side is on the x -axis and is negative. Use the Pythagorean Theorem to find the length of the hypotenuse:

(−4 ) 2 + ( 3 ) 2 = c 2 16 + 9 = c 2 25 = c 2 c = 5    

Now we can draw a triangle similar to the one shown in [link] .

Diagram of a triangle in the x,y-plane. The vertices are at the origin, (-4,0), and (-4,3). The angle at the origin is theta. The angle formed by the side (-4,3) to (-4,0) forms a right angle with the x axis. The hypotenuse across from the right angle is length 5.
  1. Let’s begin by writing the double-angle formula for sine.
    sin ( 2 θ ) = 2   sin   θ   cos   θ

    We see that we to need to find   sin   θ   and   cos   θ .   Based on [link] , we see that the hypotenuse equals 5, so   sin   θ = 3 5 , and   cos   θ = − 4 5 .   Substitute these values into the equation, and simplify.

    Thus,

    sin ( 2 θ ) = 2 ( 3 5 ) ( − 4 5 )              = − 24 25
  2. Write the double-angle formula for cosine.
    cos ( 2 θ ) = cos 2 θ − sin 2 θ

    Again, substitute the values of the sine and cosine into the equation, and simplify.

    cos ( 2 θ ) = ( − 4 5 ) 2 − ( 3 5 ) 2              = 16 25 − 9 25              = 7 25
  3. Write the double-angle formula for tangent.
    tan ( 2 θ ) = 2   tan   θ 1 − tan 2 θ

    In this formula, we need the tangent, which we were given as   tan   θ = − 3 4 .   Substitute this value into the equation, and simplify.

    tan ( 2 θ ) = 2 ( − 3 4 ) 1 − ( − 3 4 ) 2             = − 3 2 1 − 9 16             = − 3 2 ( 16 7 )             = − 24 7
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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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