Fine the length of segment x.

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:

Trapezoid Area =

[ ( Sum of the rules ) × ( height ) ] ÷ 2

_________________________________

Area = [ ( 2 + 3 ) × 3 ] ÷ 2

Area = [ 5 × 3 ] ÷ 2

Area = 15 ÷ 2

Area = 7.5 m^2

Answer:

X x X=x²

X x -7=-7x

4 x X=4x

4 x -7=-28

-7x + 4x=-3x

x² + -3x -28

hope that helped :)

Step-by-step explanation:

Answer:

okie we will try our best

Answer:

1. Not equivalent

2. Not equivalent

3. Equivalent

4. Equivalent

5. Equivalent

Step-by-step explanation:

I will prove the ones that are equivalent, since the ones that aren't are pretty straightforward.

2(3h + 7) = 6h + 14

3. 6h + 7 + 7 = 6h + 14

4. 3h + 3h + 14 = 6h + 14

5. (3h + 7) + (3h + 7) = 6h + 14

1000000

Step-by-step explanation:

son mil ovhomil

Answer:

4,261

Step-by-step explanation:

after 23 hours, the number of bacteria =

4,000.

so, after 26 hours = 4,000+ (3/23 ×4000)

= 4,000 + 261 = 4,261

Step-by-step explanation:

each week kyle saves $5.00

Answer:

The opportunity cost of playing an hour of video games is not reading 3 chapters of a book.

Step-by-step explanation:

Given that Anthony can either play 4 hours of video games or read 12 chapters of a book, to determine what is Anthony's opportunity cost of playing one hour of video games, the following calculation must be performed:

4 = 12

1 = X

1 x 12/4 = X

12/4 = X

3 = X

Therefore, the opportunity cost of playing an hour of video games is not reading 3 chapters of a book.

Step-by-step explanation:

y=x-2....eqn1

x+y=12.....eqn2

now,

from eqn 1

y= x-2

again, putting value of y in eqn 2

in eqn 1

hope it helps..

Answer:

I think B

Step-by-step explanation:

Not sure but seems like so. Hope I helped U :)

Answer:

Step-by-step explanation:

Add all 4 sides to get perimeter so the first one is 9+9+2+2=22cm

the area is length x width so the first one is 9 x 2 =18cm ^2

(a) The marginal distribution of X is

Pr(X = x) = ∑ Pr(X = x, Y = y)

… = 0.0625 + 0.0625 + 0.0625 + 0.0625

… = 0.25

That is, the first equality follows from the law of total probability, with the sum taken over y from {0, 5, 10, 15}. Each probability Pr(X = x, Y = y) is given in the table to be 0.0625.

Similarly, the marginal distribution of Y is

Pr(Y = y) = 0.25

(b) Yes, they're independent because

Pr(X = x, Y = y) = 0.0625,

and

Pr(X = x) Pr(Y = y) = 0.25 • 0.25 = 0.0625.

(c) The mean of X is

E[X] = ∑ x Pr(X = x)

… = 0.25 ∑ x

… = 0.25 (0 + 5 + 10 + 15)

… = 7.5

and you would find the same mean for Y,

E[Y] = 7.5

The variance of X is

V[X] = E[X^2] - E[X]^2

… = (∑ x^2 Pr(X = x)) - 7.5^2

… = 0.25 (∑ x^2) - 56.25

… = 0.25 (0^2 + 5^2 + 10^2 + 15^2) - 56.25

… = 31.25

and similarly,

V[Y] = 31.25

(each sum is taken with x and y from {0, 5, 10, 15})

9514 1404 393

Answer:

  (a)  6

Step-by-step explanation:

The median is the middle number when they are sorted into order. If one of the numbers is 10, the remaining number must be any number less than 7. Of the offered choices, the only one less than 7 is 6.

__

The three numbers are 6, 7, 10.

Answer:

if what the person draw wot the stuffed animal there is a probability for the winner to win stuffed animal

-√25    rational

[tex]\frac{14}{19}[/tex]          irrational

9π        irrational

2√2     irrational

87.39... rational

Answer:

The solutions are the following:

hope this helps!! :) --Siveth

Answer:

c

Step-by-step explanation:

right on plato

Answer:

I don’t know

Step-by-step explanation:

You didn’t attach a photo lol

Answer:

Step-by-step explanation:

Hope it helps u!!

Answer:

b 112.80

Step-by-step explanation:

took test on edge

The interest that accrues on the initial down payment at 30% interest rate is  $408 .

The term interest refers to money that accrues on a principal sum. It is commonly employed in the parlance of borrowing money.

Since it is a compound interest;

A = P(1 + r/n)^nt

Where;

A = amount

P = principal

r = rate

n = number of times compounded per year

t = total time taken

A = 120(1 + 0.025)^12 × 5

A = $528

A = P + I

I = A - P

I = $528  - $120

I = $408

Learn more about compound interest: https://brainly.com/question/25857212

Answer:

See explanation. I attached a photo to help visualize and make it easier to understand, but tell me if handwriting was fine :)

Also, C) 436 in² is correct :)

Step-by-step explanation:

the answer is C

Step-by-step explanation:

if we take the 15 from 28

13 will be left

so 13×22=286

and 15×5=75

15×5=75

150+286=436

hopefully its helpful

Answer:

16.73 I THINK

Step-by-step explanation:

my guess is n1 is relating to the 4.9, and if it then you would do the Pythagorean theorem. because the degrees are there to let you know it's a right angle and you are able to use the theorem..... so -

4.9^2 + 4.9^2 =C^2

48.02 =C

√48 = 6.93 (rounded to nearest hundredth)

then you add them all together

4.9+4.9+6.93=16.73

Answer:

False because 1/2 would be half the number 2/2 or 1 would be the whole number so 3/2 would be a whole and half or 1 1/2

Answer:

false

if the number is X

3/2 of X= X 1/2

[tex]x \frac{1}{2} > x[/tex]

Answer:

[tex]\frac{5}{36}[/tex]

Step-by-step explanation:

probability of desired outcome:

Number of desired outcomes/number of total outcomes

To find the numerator just count the number of times the condition is satisfied

I get 5

1,5

2,4

3,3

4,2

5,1

The number of total outcomes is just 6² or 36

So the final answer is just 5/36

Answer:

08-Oct-2009 — If a spherical balloon has a volume of 972 pi cubic centimeters, what is ... is 4.pi.r^2 (its area remember not volume) as 4/3.pi.r^3=972pi r=9 ... and it is good to know the squares of the next 10 numbers.

Answer:

I have this same question on my math test.

Since the radius is 9, the diameter is 18.

The shape is being dilated by 2/3, so the new radius is 6.

Volume formula: V = 4/3πr^3

So now we plug in the values.

V = 4/3 x π x (6)^3

V = 4/3 x π x 216

V = 288π

If the answers do not have the pi symbol present, then it has been multiplied into the answer.

288 x 3.14 = 904.32

The new volume of the sphere is 904.32 in^3

Answer:

(6, 7)

Step-by-step explanation:

A rectangle means that opposite sides are parallel to each other and that opposite sides are equal in measurements.

If we look at the opposite side starting at (2, 1), we can count how long the other side should be. Counting the next point of the rectangle is (6, 1) which is a difference of 4. This means if we count from (2, 7) 4 spaces to the right, we will find our missing vertex.

(2 + 4, 7) = (6, 7)

Answer:

Cos(θ) = 8 / 17

Step-by-step explanation:

Cos = Adjacent / Hypotenuse

Side that is adjacent to θ = 8

hypotenuse of the triangle = 17

Cos(θ) = 8 / 17

Hope this helps!