Would someone be able to assist me with this problem?

Find all solutions of each equation on the interval 0 ≤ x < 2π.

Korvikt [17]

Answer:

$x = 0$ or $x = \pi$ .

Step-by-step explanation:

How are tangents and secants related to sines and cosines?

$\displaystyle \tan{x} = \frac{\sin{x}}{\cos{x}}$ .

$\displaystyle \sec{x} = \frac{1}{\cos{x}}$ .

Sticking to either cosine or sine might help simplify the calculation. By the Pythagorean Theorem, $\sin^{2}{x} = 1 - \cos^{2}{x}$ . Therefore, for the square of tangents,

$\displaystyle \tan^{2}{x} = \frac{\sin^{2}{x}}{\cos^{2}{x}} = \frac{1 - \cos^{2}{x}}{\cos^{2}{x}}$ .

This equation will thus become:

$\displaystyle \frac{1 - \cos^{2}{x}}{\cos^{2}{x}} \cdot \frac{1}{\cos^{2}{x}} + \frac{2}{\cos^{2}{x}} - \frac{1 - \cos^{2}{x}}{\cos^{2}{x}} = 2$ .

To simplify the calculations, replace all $\cos^{2}{x}$ with another variable. For example, let $u = \cos^{2}{x}$ . Keep in mind that $0 \le \cos^{2}{x} \le 1 \implies 0 \le u \le 1$ .

$\displaystyle \frac{1 - u}{u^{2}} + \frac{2}{u} - \frac{1 - u}{u} = 2$ .

$\displaystyle \frac{(1 - u) + u - u \cdot (1- u)}{u^{2}} = 2$ .

Solve this equation for $u$ :

$\displaystyle \frac{u^{2} + 1}{u^{2}} = 2$ .

$u^{2} + 1 = 2 u^{2}$ .

$u^{2} = 1$ .

Given that $0 \le u \le 1$ , $u = 1$ is the only possible solution.

$\cos^{2}{x} = 1$ ,

$x = k \pi$ , where $k\in \mathbb{Z}$ (i.e., $k$ is an integer.)

Given that $0 \le x < 2\pi$ ,

$0 \le k$ .

$k = 0$ or $k = 1$ . Accordingly,

$x = 0$ or $x = \pi$ .

8 0

2 years ago

Read 2 more answers

The equation y=2x+3 represents the cost y (in dollars) of mailing a package that weighs x pounds. a. Graph the equation. b. Usin

FinnZ [79.3K]

Answer:

B. Slightly more than $5

Step-by-step explanation:

y = 2x + 3

Firstly we check variables to make sure we understand.

y = dollars

x = pounds

On the scale, it says 1.126 lb. Which means 1.126 pounds.

And so, x = 1.126

y = 2x + 3

y = 2(1.126) + 3

y = 2.252 + 3

y = 2.252

The graph for part a and park b. It is underneath. I wrote 1.1, because it is rounded down from 1.126 lb, which is part b. The one with one line and no dots is part a.

4 0

2 years ago

Please help me it's urgent

uysha [10]

Answer: go to the website

Step-by-step explanation:

7 0

1 year ago

Three polynomials are factored below, but some coefficients and constants are missing. All of the missing values of a, b, c, and

mariarad [96]

9514 1404 393

Answer:

(a, b, c, d) = (1, -4, 1, -2)
(a, b, c, d) = (1, -4, 1, 2)
(a, b, c, d) = (1, -4, 2, 6)

Step-by-step explanation:

Once you remove the common factor from the terms, you are looking for factors of the remaining constant term that have a sum equal to the coefficient of the linear term. These factors are the constants in the binomial factors.

__

1. 8 = (-4)(-2) ⇒ x^2 -6x +8 = (x -4)(x -2)

(a, b, c, d) = (1, -4, 1, -2)

__

2. -8 = (-4)(+2) ⇒ 3x^3 -6x^2 -24x = 3x(x -4)(x +2)

(a, b, c, d) = (1, -4, 1, 2)

__

3. -12 = (-4)(3) ⇒ 2x^2 -2x -24 = 2(x -4)(x +3) = (x -4)(2x +6)

(a, b, c, d) = (1, -4, 2, 6) or (2, -8, 1, 3)

3 0

3 years ago

If Brad had a hamster that weight 7.89 grams and his friend had a hamster that weight3x's more grams. How many grams altogether

Andreyy89

His friend's would weigh 23.67 grams.Altogether it would be 31.56 grams.

7 0

3 years ago