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If sin θ = cot θ, then, what is the value of cos θ + 2 cos^2 θ + 2 cos^3 θ + cos^ 4 θ ?

VARVARA [1.3K]

Notice that

cos(θ) + 2 cos²(θ) + 2 cos³(θ) + cos⁴(θ)

= cos(θ) (1 + 2 cos(θ) + 2 cos²(θ) + cos³(θ))

= cos(θ) ([1 + cos(θ)] + [cos(θ) + cos²(θ)] + [cos²(θ) + cos³(θ)])

= cos(θ) ([1 + cos(θ)] + [cos(θ) (1 + cos(θ))] + [cos²(θ) (1 + cos(θ))])

= cos(θ) (1 + cos(θ)) (1 + cos(θ) + cos²(θ))

Given that sin(θ) = cot(θ), by definition of cotangent this tells us that

sin(θ) = cos(θ)/sin(θ) ⇒ cos(θ) = sin²(θ)

and by the Pythagorean identity

cos²(θ) + sin²(θ) = 1

it follows that

cos(θ) = sin²(θ) = 1 - cos²(θ)

Substituting these results into the factorization above gives

cos(θ) (1 + cos(θ)) (1 + cos(θ) + cos²(θ))

= cos(θ) (1 + cos(θ)) (1 + [1 - cos²(θ)] + cos²(θ))

= 2 cos(θ) (1 + cos(θ))

= 2 sin²(θ) (1 + cos(θ))

= 2 (1 - cos²(θ)) (1 + cos(θ))

= 2 (1 + cos(θ) - cos²(θ) - cos³(θ))

= 2 (cos(θ) + cos(θ) - cos³(θ))

= 2 (2 cos(θ) - cos³(θ))

= 2 cos(θ) (2 - cos²(θ))

= 2 cos(θ) (1 + cos(θ))

= 2 (cos(θ) + cos²(θ))

= 2 (1 - cos²(θ) + cos²(θ))

= 2

7 0

3 years ago

Will give brainliest!

Kruka [31]

Correct answer: A, 2

5 0

3 years ago

Read 2 more answers

Anya was rowing up the river and noticed it takes longer to row upstream than downstream. She knows that in still water she can

Alenkinab [10]

If the speed of boat in still water iss 9 mph and takes 1.5 miles to go and come 4 miles uptream and downward then the current of the river is 9.24 mph.

Given that the speed in still water be 9 mph, time to travel 4 miles upstream and downward is 1.5 hours.

We are required to find the speed of current of the river.

let the speed of current be c mph.

Effective speed in downstream=9+c

Effective speed in upstream=9-c

Total time=1.5 hours.

Total distance=4 miles.

Time for upstream stream+ time for downward stream=1.5 hours

2/(9+c)+2/(9-c)=1.5

18-2c+18+2c=1.5(9+c)(9-c)

36=1.5(81- $c^{2}$ )

36=121.5-1.5 $c^{2}$

1.5 $c^{2}$ =121.5-36

$c^{2}$ =85.5

c=9.24 mph.

Hence the speed of the current of the river is 9.24 mph.

Learn more about speed at brainly.com/question/4931057

#SPJ4

4 0

2 years ago

Is 8 a solution to the equation below? Yes or No?<br><br><br> 3(x - 2) + 4 = 22

Simora [160]

To see if 8 is the solution to the equation, 3 (x - 2) + 4 = 22, you must replace x with 8 and solve using the rules of PEMDAS (Parentheses, Exponent, Multiplication, Division, Addition, Subtraction) REMEMBER: IF NOT APPLICABLE TO THE EQUATION YOU MAY SKIP THAT STEP IN PEMDAS . If the sides are equal to each other then 8 IS a solution.

3(8 - 2) + 4 = 22

Parentheses...

3(8 - 2) + 4 = 22

8 - 2 <<<In the parentheses so you must solve first

6

so...

3(6) + 4 = 22

Multiplication...

3(6) + 4 = 22

3 * 6

18

so..

18 + 4 = 22

Addition...

18 + 4 = 22

18 + 4

22

so...

22 = 22

Are they equal to each other? YES!!!

8 is a solution to the equation 3 (x - 2) + 4 = 22

Hope this helped!

~Just a girl in love with Shawn Mendes

7 0

3 years ago

Are these rates equivalent, 96 words typed for 3 minutes; 160 words typed for 5 minutes

monitta

<h2>Greetings</h2>

Answer:

Yes, they are.

Step-by-step explanation:

Lets simplify both rates to the rate typer per minute.

(m is minutes)

3m = 96

Divide both sides by 3:

$\frac{3m}{3} = \frac{96}{3}$

m = 32

So the rate is 32 words per minute.

5m = 160

Divide both sides by 5:

$\frac{5m}{5} = \frac{160}{5}$

m = 32

<h3>So because the rates per minute are the same, they are equivalent.</h3>
<h2>Hope this helps!</h2>

3 0

3 years ago