Find all angles, 0, degrees, is less than or equal to, theta, is less than, 360, degrees, comma0 
∘
 ≤θ<360 
∘
 , that satisfy the equation below, to the nearest tenth of a degree.
cosine, left parenthesis, theta, right parenthesis, equals, start fraction, square root of, 42, end square root, divided by, 8, end fraction
cos(θ)= 
8
42
​
 
​


We are given the equation:

\[
\cos(\theta) = \frac{\sqrt{42}}{8}
\]

### Step 1: Simplify the right-hand side

First, let's approximate \(\frac{\sqrt{42}}{8}\):

\[
\sqrt{42} \approx 6.4807
\]

Thus,

\[
\frac{\sqrt{42}}{8} \approx \frac{6.4807}{8} \approx 0.8101
\]

So the equation becomes:

\[
\cos(\theta) \approx 0.8101
\]

### Step 2: Solve for \(\theta\)

To find \(\theta\), we use the inverse cosine (arccos) function:

\[
\theta_1 = \cos^{-1}(0.8101)
\]

Using a calculator:

\[
\theta_1 \approx 35.7^\circ
\]

### Step 3: Find all possible solutions

The cosine function is positive in both the first and fourth quadrants. Therefore, the two angles that satisfy \(\cos(\theta) \approx 0.8101\) are:

1. \(\theta_1 \approx 35.7^\circ\) (first quadrant)
2. \(\theta_2 = 360^\circ - 35.7^\circ \approx 324.3^\circ\) (fourth quadrant)

### Final answer:

The angles that satisfy \(\cos(\theta) = \frac{\sqrt{42}}{8}\) in the range \(0^\circ \leq \theta < 360^\circ\) are approximately:

\[
\theta = 35.7^\circ \text{ and } 324.3^\circ
\]

Would you like further details or have any questions?

### Related Questions:
1. How do you compute inverse trigonometric functions using a calculator?
2. What is the significance of the cosine function being positive or negative in different quadrants?
3. How do we determine all solutions for trigonometric equations within a given range?
4. Why is the cosine function periodic, and how does it affect the number of solutions?
5. Can this problem be solved graphically? How would the cosine graph look for this equation?

### Tip:
Always remember that trigonometric functions are periodic, so multiple solutions might exist within a given range!

Find all angles, 0° ≤ θ < 360°, that satisfy the equation: cos(θ) = sqrt(42)/8

Learn how to solve the trigonometric equation cos(θ) = sqrt(42)/8 in the range 0° ≤ θ < 360°. This solution involves finding angles using the inverse cosine function and understanding the cosine function's behavior in different quadrants. Suitable for students in Grades 10-12.

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