Resources

Resources

Bytelearn - cat image with glasses

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

b) Find the general solution of
cos theta+Cos 3theta+cos 5theta=0
OR
Prove that:
tan^(-1)x-tan^(-1)y=tan^(-1)((x-y)/(1+xy))

b) Find the general solution of $\cos \theta+\operatorname{Cos} 3 \theta+\cos 5 \theta=0$ $\newline$ OR $\newline$ Prove that: $\tan ^{-1} x-\tan ^{-1} y=\tan ^{-1} \frac{x-y}{1+x y}$

View step-by-step help

View step-by-step help

Simplify rational expressions

Full solution

Q. b) Find the general solution of

\cos \theta+\operatorname{Cos} 3 \theta+\cos 5 \theta=0

\newline

OR

\newline

Prove that:

\tan ^{-1} x-\tan ^{-1} y=\tan ^{-1} \frac{x-y}{1+x y}

Apply sum-to-product identities: Use the sum-to-product identities to simplify $\cos \theta + \cos 5\theta$ . $\cos \theta + \cos 5\theta = 2 \cdot \cos\left(\frac{\theta + 5\theta}{2}\right) \cdot \cos\left(\frac{\theta - 5\theta}{2}\right)$ = $2 \cdot \cos(3\theta) \cdot \cos(-2\theta)$ = $2 \cdot \cos(3\theta) \cdot \cos(2\theta)$
Factor out $\cos(3\theta)$ : Now we have $2 \times \cos(3\theta) \times \cos(2\theta) + \cos(3\theta)$ . Factor out $\cos(3\theta)$ . $\cos(3\theta) \times (2 \times \cos(2\theta) + 1)$
Find general solution: Set the equation equal to zero to find the general solution. $\cos(3\theta) \cdot (2 \cdot \cos(2\theta) + 1) = 0$
Solve for $\cos(3\theta)$ : Solve for when each factor is equal to zero. $\newline$ For $\cos(3\theta) = 0$ , the general solution is: $\newline$ $3\theta = (2n + 1) \cdot \frac{\pi}{2}$ , where $n$ is an integer. $\newline$ $\theta = (2n + 1) \cdot \frac{\pi}{6}$
Solve for $\cos(2\theta)$ : For $2 \cdot \cos(2\theta) + 1 = 0$ , solve for $\cos(2\theta)$ . $\newline$ $\cos(2\theta) = -\frac{1}{2}$ $\newline$ $2\theta = \pm (2n + 1) \cdot \frac{\pi}{3}$ , where $n$ is an integer. $\newline$ $\theta = \pm (2n + 1) \cdot \frac{\pi}{6}$
Combine solutions: Combine the solutions for $\theta$ . $\theta = (2n + 1) \times \frac{\pi}{6}$ and $\theta = \pm (2n + 1) \times \frac{\pi}{6}$

More problems from Simplify rational expressions

Question

Evaluate the expression for

t = -4.4

\newline

Write your answer in simplest form.

\newline

` frac{t - 4}{t - 2} =` ____

Posted 1 month ago

Question

Simplify. Write your answer using whole numbers and variables.

\newline

\frac{g}{3g^2 - 4g}

Posted 2 months ago

Question

Simplify. Express your answer as a single fraction in simplest form.

\frac{3}{2} - \frac{5p^3}{3}

Posted 2 months ago

Question

Simplify. Express your answer as a single fraction in simplest form.

\newline

\frac{1 + \frac{v^5w}{3}}{1}

\newline

Posted 2 months ago

Question

h

\newline

\frac{-5}{h - 2} = \frac{-4}{h - 3}

\newline

1

2

\newline

h =

h =

Posted 1 month ago

Question

What is the equation of the vertical asymptote of

y = \frac{1}{x + 9}

Posted 2 months ago

Question

What is the excluded value for

y = \frac{1}{x - 9} - 9

\newline

Posted 2 months ago

Question

Simplify. Write your answer using whole numbers and variables.

\newline

\frac{9k + 4}{9k^2 + 4k}

\newline

Posted 2 months ago

Question

Divide. Write your answer in simplest form.

\newline

\frac{g- 1}{10g^3 + 25g^2} \div (g- 1)

\newline

Posted 2 months ago

Question

Simplify. Express your answer as a single fraction in simplest form.

\newline

\frac{5 - \frac{3}{t^3}}{1}

\newline

Posted 2 months ago