What is the domain of f? <br> HELP PLEASE

BRAINLIESTTT ASAP! PLEASE HELP ME :)

Gre4nikov [31]

Answer:

θ = 0.896 and 2.246.

Step-by-step explanation:

We will use the identity

sin²θ + cos²θ = 1

and the double angle formula

cos2θ = cos²θ - sin²θ = (1 - sin²θ) - sin²θ = 1 - 2sin²θ

1. Convert the equation to a more solvable form.

$\begin{array}{rcll}\sin\theta + 1 & = & \cos2\theta &\\\sin\theta + 1& = & 1-2\sin^{2}\theta & \text{Used the double angle formula}\\f(\theta) = 2\sin^{2}\theta + \sin\theta -2 & = & 0 & \text{Moved terms to left-hand side}\\2u^2 + u - 2 & = & 0 & \text{Substituted u for }\sin\theta\\\end{array}$

2. Solve the quadratic

a = 2; b = 1; c = -2

(a) Set up the equation.

$u =\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}\\\\=\dfrac{-1\pm\sqrt{(1^2 - 4\times 2 \times (-2)}}{2\times2}\\\\=\dfrac{-1\pm\sqrt{1 + 16}}{4}\\\\=\dfrac{-1\pm\sqrt{17}}{4}\\$

(b) Evaluate the roots

$\begin{array}{rcl}u = \dfrac{-1 - 4.123}{4} & \qquad & u = \dfrac{-1 + 4.123}{4}\\\\u = -\dfrac{5.123}{4} & \qquad & u =\dfrac{3.123}{4}\\\\u = -1.281& \qquad & u =0.7808\\\end{array}$

3. Solve for θ

(a) The negative root

u = sinθ = -1.281

sinθ cannot be less than -1. No solution.

(b) The positive root

If sin(x) = a, then

x = arcsin(x) + 2πn and

x = π - arcsin(a) + 2πn

(i) Case 1

u = sinθ = 0.7808

θ = arcsin(0.7808) + 2πn

θ = 0.8959 ± 2πn

If n = 0,

θ = 0.8959

(ii) Case 2

θ = π - 0.8959 + 2πn

If n = 0,

θ = π - 0.8959 = 2.246

The solutions are θ = 0.8959 and θ = 2.246

The diagram below shows the graph of f(θ) with its roots at 0.896 and 2.246.

5 0

3 years ago

Simplify.<br> -3(u + 3) + 4(4u + 2)

sergeinik [125]

Hello!

So, here you want to do the distributive property.

Start on the left side of the + sign...

-3*u = -3u

-3*3 = -9

So we now we have:

(-3u + -9) + 4(4u + 2)

Now we focus on the right side of the + sign...

4*4u=16u

4*2=8

So now we have:

(-3u + -9) + (16u + 8)

Now you want to combine like terms. Start by adding terms with the same variable. So, we need to add -3u and 16u

<em>-3u + 16u =</em> 13u

Now we have:

13u + -9 + 8

Now add the constants, -9 + 8

-9 + 8 = -1

So, the simplified expression would be:

13u - 1 or 13u + -1

6 0

3 years ago

Read 2 more answers

A taxi driver uses the formula 3.00 +1.5m to calculate the cost of a taxi

irakobra [83]

Answer:

24$

Step-by-step explanation:

12m/1.5m=8 8*3=24

4 0

3 years ago

Write the equation of a circle in standard form with center ( -2,4) and radius 6. Show your work.

grigory [225]

Answer:

( x + 2 )² + ( y - 4 )² = 36

Step-by-step explanation:

Standard equation of a circle - ( x - h )² + ( y - k )² = r²

where r = radius and center is at ( h , k )

The given center is ( -2 , 4 )

Thus h = -2 and k = 4

and the given radius is 6

Thus, r = 6

that being said, we plug in the given values into the standard equation

( x - ( - 2 )² + ( y - 4 )² = 6²

Finally we simplify

In - ( -2 ) the two negative signs cancel out and it turns into + 2

6² = 36

The final equation of the circle is ( x + 2 )² + ( y - 4 )² = 36

8 0

2 years ago

What is the ratio of the area of the inner square to the area of the outer square?

Setler [38]

Answer:

Step-by-step explanation:

If we are looking for the ratio of the area of the inner square to the area of the outer square, that means that we need the areas of each of these squares, and we need to find the areas without any numbers. But that's ok; the answer they want is not a number answer. The answer will have a's and b's in it instead of numbers.

First the area of the inner square. Here we go:

Look at the triangle in the lower left corner of this coordinate plane. It is a right triangle. The height of it is b. That's because the height is a "y" thing and the y-coordinates of each of those sets of coordinates is b and 0. The height is then b - 0 = b.

The length of the base is a - b. That's because the length is an "x" thing and the x-coordinates of each of those sets of coordinates is (a - b) and 0. The length is then a - b - 0 = a - b.

Now we need the length of the hypotenuse which also serves as one of the sides of the inner square. Using Pythagorean's Theorem, we can find the length of the hypotenuse, which I will label as "?":

$?^2=b^2+(a-b)^2$ and

$?^2=b^2+a^2-2ab+b^2$ and

$?^2=a^2-2ab+2b^2$ so

?, the length of the hypotenuse, is

$?=\sqrt{a^2-2ab+2b^2}$ and now we can use that to find the area of the inner square. The formula for a square's area is

$A=s^2$ so

$A=(\sqrt{a^_2}-2ab+2b^2})^2$ which gives us finally:

$A=a^2-2ab+2b^2$ **

Now for the outer square. Those blue triangles you see are all congruent. We can use the side lengths for the triangles we found above to find the length of a side of the outer square. One side of the outer square is made up of one base length of these triangles and one height. We found the base length to be (a - b) and the height to be b; therefore, the length of one side of the outer square is b + (a - b) which is just "a". That's is, just a length of "a". The area is found by multiplying this side length by itself, so the area of the outer square is

A = a²

The ratio of the area of the inner to the outer is:

$\frac{A_i}{A_o}:\frac{a^2-2ab+2b^2}{a^2}$ and that does not reduce.

7 0

3 years ago

What is the domain of f? HELP PLEASE