<< Chapter < Page 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

x exposant 4 + 4 x exposant 3 + 8 exposant 2 + 4 x + 1 = 0
HERVE Reply
x exposent4+4x exposent3+8x exposent2+4x+1=0
HERVE
How can I solve for a domain and a codomains in a given function?
Oliver Reply
ranges
EDWIN
Thank you I mean range sir.
Oliver
proof for set theory
Kwesi Reply
don't you know?
Inkoom
find to nearest one decimal place of centimeter the length of an arc of circle of radius length 12.5cm and subtending of centeral angle 1.6rad
Martina Reply
factoring polynomial
Noven Reply
what's your topic about?
Shin Reply
find general solution of the Tanx=-1/root3,secx=2/root3
Nani Reply
find general solution of the following equation
Nani
the value of 2 sin square 60 Cos 60
Sanjay Reply
0.75
Lynne
0.75
Inkoom
when can I use sin, cos tan in a giving question
duru Reply
depending on the question
Nicholas
I am a carpenter and I have to cut and assemble a conventional roof line for a new home. The dimensions are: width 30'6" length 40'6". I want a 6 and 12 pitch. The roof is a full hip construction. Give me the L,W and height of rafters for the hip, hip jacks also the length of common jacks.
John
I want to learn the calculations
Koru Reply
where can I get indices
Kojo Reply
I need matrices
Nasasira
hi
Raihany
Hi
Solomon
need help
Raihany
maybe provide us videos
Nasasira
about complex fraction
Raihany
Hello
Cromwell
a
Amie
What do you mean by a
Cromwell
nothing. I accidentally press it
Amie
you guys know any app with matrices?
Khay
Ok
Cromwell
Solve the x? x=18+(24-3)=72
Leizel Reply
x-39=72 x=111
Suraj
Solve the formula for the indicated variable P=b+4a+2c, for b
Deadra Reply
Need help with this question please
Deadra
b=-4ac-2c+P
Denisse
b=p-4a-2c
Suddhen
b= p - 4a - 2c
Snr
p=2(2a+C)+b
Suraj
b=p-2(2a+c)
Tapiwa
P=4a+b+2C
COLEMAN
b=P-4a-2c
COLEMAN
like Deadra, show me the step by step order of operation to alive for b
John
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.
Kaitlyn Reply
The sequence is {1,-1,1-1.....} has
amit Reply
Practice Key Terms 3

Get the best Algebra and trigonometry course in your pocket!





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