Fractions and Decimals

Discrete Mathematics

Differential Equations

Solved on Dec 18, 2023

Solve for the acute angle $\theta$ where $3 \cos \theta = \frac{1}{9}$ .

STEP 1

Assumptions
1. We are solving for an acute angle $\theta$ , which means $0 < \theta < \frac{\pi}{2}$ .
2. The equation to solve is $3 \cos \theta = \frac{1}{9}$ .
3. We will use the inverse cosine function to find the value of $\theta$ .

STEP 2

First, we need to isolate $\cos \theta$ by dividing both sides of the equation by 3.
$\cos \theta = \frac{1}{9} \div 3$

STEP 3

Calculate the right-hand side of the equation.
$\cos \theta = \frac{1}{27}$

STEP 4

Now, we will use the inverse cosine function to find the value of $\theta$ . The inverse cosine function, denoted as $\cos^{-1}$ , gives us the angle whose cosine is the number provided.
$\theta = \cos^{-1}\left(\frac{1}{27}\right)$

STEP 5

Use a calculator to find the value of $\theta$ . Make sure the calculator is set to the correct angle unit (radians or degrees) as needed. Since we are looking for an acute angle, we will assume the angle is measured in degrees.
$\theta \approx \cos^{-1}\left(\frac{1}{27}\right)$

STEP 6

Calculate the value of $\theta$ in degrees.
$\theta \approx \cos^{-1}\left(\frac{1}{27}\right) \approx 88.85^\circ$
Since $\theta$ is an acute angle and the calculated value is less than $90^\circ$ , this is the solution.
$\theta \approx 88.85^\circ$

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