Use inverse trig functions to find the smallest two positive solutions of cos(2x+0.5) = 0.5978 on the interval [0,2pi)

To find the smallest two positive solutions of the equation \( \cos(2x + 0.5) = 0.5978 \) on the interval \( [0, 2\pi) \), we'll follow these steps:

### Step 1: Use the inverse cosine function
We know that \( \cos(2x + 0.5) = 0.5978 \). To find the angle that gives this cosine value, we'll take the inverse cosine:

\[
2x + 0.5 = \cos^{-1}(0.5978)
\]

Using a calculator, we find:

\[
2x + 0.5 = \cos^{-1}(0.5978) \approx 0.9336 \text{ radians}
\]

### Step 2: General solutions for the cosine equation
The cosine function has a period of \( 2\pi \), so the general solution for \( \cos(\theta) = 0.5978 \) is:

\[
2x + 0.5 = 0.9336 + 2k\pi \quad \text{or} \quad 2x + 0.5 = -0.9336 + 2k\pi
\]

where \( k \) is any integer.

### Step 3: Solve for \( x \)
For the first case:

\[
2x + 0.5 = 0.9336 + 2k\pi
\]

Subtract 0.5 from both sides:

\[
2x = 0.4336 + 2k\pi
\]

Divide by 2:

\[
x = 0.2168 + k\pi
\]

For the second case:

\[
2x + 0.5 = -0.9336 + 2k\pi
\]

Subtract 0.5 from both sides:

\[
2x = -1.4336 + 2k\pi
\]

Divide by 2:

\[
x = -0.7168 + k\pi
\]

### Step 4: Find the smallest two positive solutions
We need to find the smallest two positive values of \( x \) in the interval \( [0, 2\pi) \).

For \( x = 0.2168 + k\pi \):

- For \( k = 0 \), \( x = 0.2168 \).
- For \( k = 1 \), \( x = 0.2168 + \pi \approx 0.2168 + 3.1416 \approx 3.3584 \).

For \( x = -0.7168 + k\pi \):

- For \( k = 1 \), \( x = -0.7168 + \pi \approx 2.4248 \).
- For \( k = 2 \), \( x = -0.7168 + 2\pi \approx -0.7168 + 6.2832 \approx 5.5664 \).

The smallest two positive solutions are:

- \( x_1 \approx 0.2168 \) (from \( x = 0.2168 + k\pi \) with \( k = 0 \))
- \( x_2 \approx 2.4248 \) (from \( x = -0.7168 + k\pi \) with \( k = 1 \))

### Final Answer:
The smallest two positive solutions in the interval \( [0, 2\pi) \) are approximately:
- \( x_1 \approx 0.2168 \) radians
- \( x_2 \approx 2.4248 \) radians

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Would you like more details or have any questions?

### Related Questions:
1. How do you find the general solution for a trigonometric equation?
2. What are the properties of the cosine function?
3. How can you verify if your solutions are correct within a specific interval?
4. What are the implications of adding multiples of \( 2\pi \) to trigonometric solutions?
5. How do inverse trigonometric functions work, particularly \( \cos^{-1}(x) \)?
6. What are other methods to solve trigonometric equations besides inverse functions?
7. How do you find the period of a trigonometric function?
8. What is the significance of the principal value in inverse trigonometric functions?

### Tip:
Always check your solutions against the original equation, especially when dealing with inverse trigonometric functions, to ensure they fall within the desired interval.

Learn how to use inverse trigonometric functions to find the smallest two positive solutions of the equation \( \cos(2x + 0.5) = 0.5978 \) on the interval \( [0, 2\pi) \). This detailed solution covers trigonometric concepts, including inverse cosine function usage and periodicity. Suitable for advanced high school or college students.

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