Resources

Resources

Bytelearn - cat image with glasses

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

Find all angles,
0^(@) <= theta < 360^(@), that satisfy the equation below, to the nearest tenth of a degree.

-6cos^(2)theta-5cos theta=1
Answer:
theta=

Find all angles, $0^{\circ} \leq \theta<360^{\circ}$ , that satisfy the equation below, to the nearest tenth of a degree. $\newline$ $-6 \cos ^{2} \theta-5 \cos \theta=1$ $\newline$ Answer: $\theta=$

View step-by-step help

View step-by-step help

Find the roots of factored polynomials

Full solution

Q. Find all angles,

0^{\circ} \leq \theta<360^{\circ}

, that satisfy the equation below, to the nearest tenth of a degree.

\newline

-6 \cos ^{2} \theta-5 \cos \theta=1

\newline

Answer:

\theta=

Rewrite Equation: Let's first rewrite the equation in a more familiar quadratic form by substituting $\cos(\theta)$ with a variable, let's say ' $x$ '. So the equation becomes: $\newline$ $-6x^2 - 5x = 1$ $\newline$ Now, we need to solve for ' $x$ ' before we can find the corresponding angles for $\theta$ .
Solve Quadratic Equation: To solve the quadratic equation, we first move all terms to one side to set the equation to zero: $\newline$ $-6x^2 - 5x - 1 = 0$ $\newline$ Now, we can use the quadratic formula to find the solutions for 'x'. The quadratic formula is $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ , where $a = -6$ , $b = -5$ , and $c = -1$ .
Calculate Discriminant: Let's calculate the discriminant $(b^2 - 4ac)$ first: $\newline$ Discriminant = $(-5)^2 - 4(-6)(-1) = 25 - 24 = 1$ $\newline$ Since the discriminant is positive, we will have two real solutions for $'x'$ .
Find Solutions for x: Now, we can find the two solutions for 'x' using the quadratic formula: $\newline$ $x = \frac{-(-5) \pm \sqrt{1}}{2 \cdot -6}$ $\newline$ $x = \frac{5 \pm 1}{-12}$ $\newline$ This gives us two solutions for 'x': $\newline$ $x_1 = \frac{5 + 1}{-12} = \frac{6}{-12} = -0.5$ $\newline$ $x_2 = \frac{5 - 1}{-12} = \frac{4}{-12} = -\frac{1}{3}$
Find Angles for $x_1$ : Now we need to find the angles $\theta$ that correspond to the cosine values of $x_1$ and $x_2$ . Since we are looking for angles between $0$ and $360$ degrees, we will use the inverse cosine function and also consider the symmetry of the cosine function. $\newline$ For $x_1 = -0.5$ , we find the related angle: $\newline$ $\theta_1 = \cos^{-1}(-0.5)$
Calculate $\theta_1$ and $\theta_2$ : Using a calculator, we find: $\newline$ $\theta_1 = \cos^{-1}(-0.5) \approx 120$ degrees $\newline$ However, since cosine is positive in the fourth quadrant as well, we also have another angle: $\newline$ $\theta_2 = 360 - \theta_1 \approx 360 - 120 = 240$ degrees
Find Angles for $x_2$ : For $x_2 = -\frac{1}{3}$ , we find the related angle: $\newline$ $\theta_3 = \cos^{-1}(-\frac{1}{3})$ $\newline$ Using a calculator, we find: $\newline$ $\theta_3 \approx \cos^{-1}(-\frac{1}{3}) \approx 109.5$ degrees (to the nearest tenth) $\newline$ Again, considering the symmetry of the cosine function, we find another angle: $\newline$ $\theta_4 = 360 - \theta_3 \approx 360 - 109.5 = 250.5$ degrees (to the nearest tenth)

More problems from Find the roots of factored polynomials

Question

Find the degree of this polynomial.

\newline

`5b^{10} + 3b^3`

\newline

Posted 3 months ago

Question

\newline

(9a + 5) - (2a + 4)

\newline

Posted 3 months ago

Question

Find the product. Simplify your answer.

\newline

-3v^2(v^2 - 9)

\newline

Posted 2 months ago

Question

Find the product. Simplify your answer.

\newline

(v - 3)(4v + 1)

\newline

Posted 3 months ago

Question

Find the square. Simplify your answer.

\newline

(3y + 2)^2

\newline

Posted 3 months ago

Question

Divide. If there is a remainder, include it as a simplified fraction.

\newline

(24t^2 + 36t) \div 6t

\newline

Posted 2 months ago

Question

Is the function

q(x) = x^6 - 9

even, odd, or neither?

\newline

\newline

\text{[[even][odd][neither]]}

Posted 3 months ago

Question

Find the product. Simplify your answer.

\newline

-3q^2(-3q^2 + q)

\newline

Posted 3 months ago

Question

Find the product. Simplify your answer.

\newline

(r + 3)(4r + 2)

\newline

Posted 3 months ago

Question

Find the roots of the factored polynomial.

\newline

(x + 7)(x + 4)

\newline

Write your answer as a list of values separated by commas.

\newline

x =

Posted 3 months ago