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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.


    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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Questions & Answers

can you not take the square root of a negative number
Sharon Reply
No because a negative times a negative is a positive. No matter what you do you can never multiply the same number by itself and end with a negative
Actually you can. you get what's called an Imaginary number denoted by i which is represented on the complex plane. The reply above would be correct if we were still confined to the "real" number line.
Suppose P= {-3,1,3} Q={-3,-2-1} and R= {-2,2,3}.what is the intersection
Elaine Reply
can I get some pretty basic questions
Ama Reply
In what way does set notation relate to function notation
is precalculus needed to take caculus
Amara Reply
It depends on what you already know. Just test yourself with some precalculus questions. If you find them easy, you're good to go.
the solution doesn't seem right for this problem
Mars Reply
what is the domain of f(x)=x-4/x^2-2x-15 then
Conney Reply
x is different from -5&3
All real x except 5 and - 3
how to prroved cos⁴x-sin⁴x= cos²x-sin²x are equal
jeric Reply
Don't think that you can.
By using some imaginary no.
how do you provided cos⁴x-sin⁴x = cos²x-sin²x are equal
jeric Reply
What are the question marks for?
Someone should please solve it for me Add 2over ×+3 +y-4 over 5 simplify (×+a)with square root of two -×root 2 all over a multiply 1over ×-y{(×-y)(×+y)} over ×y
Abena Reply
For the first question, I got (3y-2)/15 Second one, I got Root 2 Third one, I got 1/(y to the fourth power) I dont if it's right cause I can barely understand the question.
Is under distribute property, inverse function, algebra and addition and multiplication function; so is a combined question
find the equation of the line if m=3, and b=-2
Ashley Reply
graph the following linear equation using intercepts method. 2x+y=4
ok, one moment
how do I post your graph for you?
it won't let me send an image?
also for the first one... y=mx+b so.... y=3x-2
y=mx+b you were already given the 'm' and 'b'. so.. y=3x-2
Please were did you get y=mx+b from
y=mx+b is the formula of a straight line. where m = the slope & b = where the line crosses the y-axis. In this case, being that the "m" and "b", are given, all you have to do is plug them into the formula to complete the equation.
thanks Tommy
0=3x-2 2=3x x=3/2 then . y=3/2X-2 I think
co ordinates for x x=0,(-2,0) x=1,(1,1) x=2,(2,4)
"7"has an open circle and "10"has a filled in circle who can I have a set builder notation
Fiston Reply
Where do the rays point?
x=-b+_Гb2-(4ac) ______________ 2a
Ahlicia Reply
I've run into this: x = r*cos(angle1 + angle2) Which expands to: x = r(cos(angle1)*cos(angle2) - sin(angle1)*sin(angle2)) The r value confuses me here, because distributing it makes: (r*cos(angle2))(cos(angle1) - (r*sin(angle2))(sin(angle1)) How does this make sense? Why does the r distribute once
Carlos Reply
so good
this is an identity when 2 adding two angles within a cosine. it's called the cosine sum formula. there is also a different formula when cosine has an angle minus another angle it's called the sum and difference formulas and they are under any list of trig identities
strategies to form the general term
consider r(a+b) = ra + rb. The a and b are the trig identity.
How can you tell what type of parent function a graph is ?
Mary Reply
generally by how the graph looks and understanding what the base parent functions look like and perform on a graph
if you have a graphed line, you can have an idea by how the directions of the line turns, i.e. negative, positive, zero
y=x will obviously be a straight line with a zero slope
y=x^2 will have a parabolic line opening to positive infinity on both sides of the y axis vice versa with y=-x^2 you'll have both ends of the parabolic line pointing downward heading to negative infinity on both sides of the y axis
y=x will be a straight line, but it will have a slope of one. Remember, if y=1 then x=1, so for every unit you rise you move over positively one unit. To get a straight line with a slope of 0, set y=1 or any integer.
yes, correction on my end, I meant slope of 1 instead of slope of 0
what is f(x)=
Karim Reply
I don't understand
Typically a function 'f' will take 'x' as input, and produce 'y' as output. As 'f(x)=y'. According to Google, "The range of a function is the complete set of all possible resulting values of the dependent variable (y, usually), after we have substituted the domain."
Sorry, I don't know where the "Â"s came from. They shouldn't be there. Just ignore them. :-)
It is the  that should not be there. It doesn't seem to show if encloses in quotation marks. "Â" or 'Â' ... Â
Now it shows, go figure?
Practice Key Terms 3

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