1. Home
  2. Algebra and trigonometry
  3. Trigonometric identities and
  4. Double-angle, half-angle, and

9.3 Double-angle, half-angle, and reduction formulas

<< Chapter < Page
  Algebra and trigonometry     Page 1 / 8
Chapter >> Page >
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 variations. The first variation is:

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

The second variation 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
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Questions & Answers

how to study physic and understand
Ewa Reply
what is conservative force with examples
Moses
what is work
Fredrick Reply
the transfer of energy by a force that causes an object to be displaced; the product of the component of the force in the direction of the displacement and the magnitude of the displacement
AI-Robot
why is it from light to gravity
Esther Reply
difference between model and theory
Esther
Is the ship moving at a constant velocity?
Kamogelo Reply
The full note of modern physics
aluet Reply
introduction to applications of nuclear physics
aluet Reply
the explanation is not in full details
Moses Reply
I need more explanation or all about kinematics
Moses
yes
zephaniah
I need more explanation or all about nuclear physics
aluet
Show that the equal masses particles emarge from collision at right angle by making explicit used of fact that momentum is a vector quantity
Muhammad Reply
yh
Isaac
A wave is described by the function D(x,t)=(1.6cm) sin[(1.2cm^-1(x+6.8cm/st] what are:a.Amplitude b. wavelength c. wave number d. frequency e. period f. velocity of speed.
Majok Reply
what is frontier of physics
Somto Reply
A body is projected upward at an angle 45° 18minutes with the horizontal with an initial speed of 40km per second. In hoe many seconds will the body reach the ground then how far from the point of projection will it strike. At what angle will the horizontal will strike
Gufraan Reply
Suppose hydrogen and oxygen are diffusing through air. A small amount of each is released simultaneously. How much time passes before the hydrogen is 1.00 s ahead of the oxygen? Such differences in arrival times are used as an analytical tool in gas chromatography.
Ezekiel Reply
please explain
Samuel
what's the definition of physics
Mobolaji Reply
what is physics
Nangun Reply
the science concerned with describing the interactions of energy, matter, space, and time; it is especially interested in what fundamental mechanisms underlie every phenomenon
AI-Robot
what is isotopes
Nangun Reply
nuclei having the same Z and different N s
AI-Robot
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply
<< Chapter < Page Page > Chapter >>
Practice Key Terms 3

Read also:

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Algebra and trigonometry' conversation and receive update notifications?

Ask