https://file.aiquickdraw.com/mathbot/file/1726799146876sp4fxbtt.jpg

The problem provided asks to find all solutions of the equation \(\cos \theta = 0.7211\) in the interval \([0, 2\pi)\). The solution should be expressed in radians and rounded to the nearest hundredth.

### Step-by-step Solution:
1. **Find the principal solution**:
   \[
   \theta = \cos^{-1}(0.7211)
   \]
   We will calculate this using the inverse cosine function.

2. **Consider the cosine function**:
   The cosine function is positive in the first and fourth quadrants. So, we will need to find two solutions in the interval \([0, 2\pi)\):
   - One solution in the first quadrant (which is the principal solution).
   - Another solution in the fourth quadrant.

3. **The general solution for cosine**:
   If \(\theta_1\) is the principal solution, the second solution is given by:
   \[
   \theta_2 = 2\pi - \theta_1
   \]

Let's now compute the exact values. I'll do that now.The solutions to the equation \(\cos \theta = 0.7211\) in the interval \([0, 2\pi)\) are:

\[
\theta_1 = 0.77 \text{ radians}, \quad \theta_2 = 5.52 \text{ radians}
\]

These are the values rounded to the nearest hundredth.

Would you like more details on any step or have any further questions?

Here are 5 related questions:
1. How do we know cosine is positive in both the first and fourth quadrants?
2. How would the solutions change if \(\cos \theta\) were negative?
3. What is the general form of solutions for \(\cos \theta = k\)?
4. How does the interval \([0, 2\pi)\) influence the number of solutions?
5. What are the solutions if we switch to degrees instead of radians?

**Tip**: When solving trigonometric equations, always check the domain (interval) to determine how many solutions are valid.

Hallar todas las soluciones de la ecuación siguiente en el intervalo [0, 2π): cosθ = 0.7211.

Solve the equation cosθ = 0.7211 in the interval [0, 2π). Learn how to find all solutions using the inverse cosine function, explore the cosine function in the first and fourth quadrants, and round the answers to the nearest hundredth.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation