use the double-angle formulas for sine and cosine to verify these identities:

tan(2theta) = 2tan(theta) / 1 - tan^2 (theta)​

sin

(

2

θ

)

=

2

(

3

5

)

(

−

4

5

)

=

−

24

25

Write the double-angle formula for cosine.

\displaystyle \cos \left(2\theta \right)={\cos }^{2}\theta -{\sin }^{2}\thetacos(2θ)=cos

​2

​​ θ−sin

​2

​​ θ

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

)Answer:

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 \displaystyle \alpha =\betaα=β. Deriving the double-angle formula for sine begins with the sum formula,

\displaystyle \sin \left(\alpha +\beta \right)=\sin \alpha \cos \beta +\cos \alpha \sin \betasin(α+β)=sinαcosβ+cosαsinβ

If we let \displaystyle \alpha =\beta =\thetaα=β=θ, then we have

Deriving the double-angle for cosine gives us three options. First, starting from the sum formula, \displaystyle \cos \left(\alpha +\beta \right)=\cos \alpha \cos \beta -\sin \alpha \sin \betacos(α+β)=cosαcosβ−sinαsinβ, and letting \displaystyle \alpha =\beta =\thetaα=β=θ, we have

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

Similarly, to derive the double-angle formula for tangent, replacing \displaystyle \alpha =\beta =\thetaα=β=θ in the sum formula gives

\displaystyle \begin{array}{c}\tan \left(\alpha +\beta \right)=\frac{\tan \alpha +\tan \beta }{1-\tan \alpha \tan \beta }\\ \tan \left(\theta +\theta \right)=\frac{\tan \theta +\tan \theta }{1-\tan \theta \tan \theta }\\ \tan \left(2\theta \right)=\frac{2\tan \theta }{1-{\tan }^{2}\theta }\end{array}

​tan(α+β)=

​1−tanαtanβ

​

​tanα+tanβ

​​

​tan(θ+θ)=

​1−tanθtanθ

​

​tanθ+tanθ

​​

​tan(2θ)=

​1−tan

​2

​​ θ

​

​2tanθ

​​

​​

A GENERAL NOTE: DOUBLE-ANGLE FORMULAS

The double-angle formulas are summarized as follows:

\displaystyle \sin \left(2\theta \right)=2\sin \theta \cos \thetasin(2θ)=2sinθcosθ

\displaystyle \tan \left(2\theta \right)=\frac{2\tan \theta }{1-{\tan }^{2}\theta }tan(2θ)=

​1−tan

​2

​​ θ

​

​2tanθ

​​

HOW TO: GIVEN THE TANGENT OF AN ANGLE AND THE QUADRANT IN WHICH IT IS LOCATED, USE THE DOUBLE-ANGLE FORMULAS TO FIND THE EXACT VALUE.

Draw a triangle to reflect the given information.

Determine the correct double-angle formula.

Substitute values into the formula based on the triangle.

Simplify.

EXAMPLE 1: USING A DOUBLE-ANGLE FORMULA TO FIND THE EXACT VALUE INVOLVING TANGENT

Given that

and \displaystyle \thetaθ is in quadrant II, find the following:

\displaystyle \sin \left(2\theta \right)sin(2θ)

\displaystyle \cos \left(2\theta \right)cos(2θ)

\displaystyle \tan \left(2\theta \right)tan(2θ)

SOLUTION

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 \displaystyle \tan \theta =-\frac{3}{4}tanθ=−

​4

​

​3

​​ , such that \displaystyle \thetaθ is in quadrant II. The tangent of an angle is equal to the opposite side over the adjacent side, and because \displaystyle \thetaθ 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:

Now we can draw a triangle similar to the one shown in Figure 2.

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.

Figure 2

Let’s begin by writing the double-angle formula for sine.

\displaystyle \sin \left(2\theta \right)=2\sin \theta \cos \thetasin(2θ)=2sinθcosθ

We see that we to need to find \displaystyle \sin \thetasinθ and \displaystyle \cos \thetacosθ. Based on Figure 2, we see that the hypotenuse equals 5, so \displaystyle \sin \theta =\frac{3}{5}sinθ=

. Substitute these values into the equation, and simplify.

Thus,

​​

Solution

EXAMPLE 2: USING THE DOUBLE-ANGLE FORMULA FOR COSINE WITHOUT EXACT VALUES

Use the double-angle formula for cosine to write \displaystyle \cos \left(6x\right)cos(6x) in terms of \displaystyle \cos \left(3x\right)cos(3x).

SOLUTION

c

Analysis of the Solution

This example illustrates that we can use the double-angle formula without having exact values. It emphasizes that the pattern is what we need to remember and that identities are true for all values in the domain of the trigonometric function.

Using Double-Angle Formulas to Verify Identities

Establishing identities using the double-angle formulas is performed using the same steps we used to derive the sum and difference formulas. Choose the more complicated side of the equation and rewrite it until it matches the other side.

EXAMPLE 3: USING THE DOUBLE-ANGLE FORMULAS TO ESTABLISH AN IDENTITY

Establish the following identity using double-angle formulas:

\displaystyle 1+\sin \left(2\theta \right)={\left(\sin \theta +\cos \theta \right)}^{2}1+sin(2θ)=(sinθ+cosθ)

​2

​​

θ

)

Step-by-step explanation:

olve the system of linear equations by graphing. =3+3 - 2x - 3 ​

PLS HELP 40 POINTS

A rectangular prism has a surface area of 8 square feet. A similar rectangular prism has a surface area of 200 square feet. How many times larger are the dimensions of the larger prism?

25
625
5
10

Given: AC Bisects Statements reasons

Help needed today due today I will give brainliest to correct answer

What number added to four and five-tenths, is twelve?

The fraction 5/8
produces a repeating decimal.
___
0.625

True or False

Which set of data has the smallest difference between the mean and the median?
O A 13, 42, 17, 26, 38
OB. 28, 7, 36, 8, 17
OC 14, 29, 38, 7, 13
OD 23, 36, 25, 38, 30

it’s a non calc question

What annual rate of interest would you have to earn on an investment of $5500 to ensure receiving $434.50 interest after one year?
PLEASE HELPPPP

Degree and Radian Measures

Convert the angle to radians. Leave as a multiple of /pi
310°

a. -9/pi/2
b. 9/pi
c. -9/pi/4
d. 9/pi/2

Please select the best answer from choices provided

It takes a racecar 47 seconds to complete one lap on a 3,200-meter track. What is the racecar's average speed during the lap? A. 150 m/s B. 68 m/s C. 15 m/s D. 79 m/s

WILL NAME BRAINLIEST

2+9n=72
how do you find what n equals?

Express in the form 1:n
Give n as a fully simplified fraction.
4:6​

Find the measure of A,B and C. ​

HELP ME PLEASEEEEEEEEEEEE

What is the volume of a sphere with a radius
of 6?
A 721
14471
2161
IN 288rt

5 + 0.55x = 10 + 0.75x pls help and explain ​

The area of a rectangle is 200 square feet. Find its length if its width is 10 feet.

List the following from least to greatest: 3, -4, 0, 1, -9

Write the equation of the line in fully simplified slope-intercept form.

(-2x^2y^5)^3 (5x^4)/10x^7y^13

PLS DUE RN



WILL GIVE




BRAINLIST

True or False
Two major composers of minimalism include Steve Reich and John Cage.
A. true
B. false

Sorry for the wrong subject

CARD #8: Henry says that the two figures will be congruent. A figure on a coordinate plane is dilated by a scale factor that is less than one. Heather says that the two figures will be similar.​

UN


2. What is the conversion from cubic metres to litres?​

an aquarium at the zoo is shaped like a cylinder. it has a height of 4 feet and a base radius of 2.5 feet. the aquarium is being filled with water at a rate of 12 gallons per minute. if one cubic foot is about 7.5 gallons, how long will it take the aquarium to fill? round to the nearest whole minute.

Guys can you help me please :( ASAP

The quotient of 9 and 3 subtracted from the product of 6 and 2

What is the range of the data Colin collected?

Question 3 (2 points)
(09.06)
For two functions, m(x) and p(x), a statement is made that m(x) = p(x) at x = 7. What is definitely true about x = 7? (2 points)

a
Both m(x) and p(x) cross the x-axis at 7.

b
Both m(x) and p(x) cross the y-axis at 7.

c
Both m(x) and p(x) have the same output value at x = 7.

d
Both m(x) and p(x) have a maximum or minimum value at x = 7.

If x=9, what is the value of y in the following equation?
2y= 2x^2/3 +6