 Use the trigonometric substitution to write the algebraic expression as a trigonometric function of 𝜃, where
0 < 𝜃 < 
𝜋
2
.
 
	4 − x2
,
  
x = 2 cos(𝜃)


To express \( \sqrt{4 - x^2} \) as a trigonometric function of \( \theta \) using the substitution \( x = 2 \cos(\theta) \), follow these steps:

### Step 1: Substitute \( x = 2 \cos(\theta) \) into the expression
We are given:

\[
x = 2 \cos(\theta)
\]

Substitute this into \( \sqrt{4 - x^2} \):

\[
\sqrt{4 - x^2} = \sqrt{4 - (2 \cos(\theta))^2}
\]

### Step 2: Simplify the expression
Now simplify the square:

\[
\sqrt{4 - (2 \cos(\theta))^2} = \sqrt{4 - 4 \cos^2(\theta)}
\]

Factor out the 4 inside the square root:

\[
\sqrt{4 - 4 \cos^2(\theta)} = \sqrt{4(1 - \cos^2(\theta))}
\]

### Step 3: Use the Pythagorean identity
Recall the Pythagorean identity:

\[
1 - \cos^2(\theta) = \sin^2(\theta)
\]

Substitute this into the expression:

\[
\sqrt{4(1 - \cos^2(\theta))} = \sqrt{4 \sin^2(\theta)}
\]

### Step 4: Simplify the square root
Take the square root:

\[
\sqrt{4 \sin^2(\theta)} = 2 \sin(\theta)
\]

### Final Answer:
Thus, the expression \( \sqrt{4 - x^2} \) is equivalent to:

\[
\sqrt{4 - x^2} = 2 \sin(\theta)
\]

### Do you want any further details or explanations on any of the steps?

### Related Questions:
1. How would you perform trigonometric substitution for \( \sqrt{x^2 - 4} \) with \( x = 2 \sec(\theta) \)?
2. What is the general strategy for choosing the appropriate trigonometric substitution?
3. How can trigonometric identities simplify integrals involving radicals?
4. Can you apply this substitution in evaluating integrals? If so, how?
5. What happens if \( \theta \) is in a different range, like \( \frac{\pi}{2} < \theta < \pi \)?

### Tip:
Trigonometric identities like \( \sin^2(\theta) + \cos^2(\theta) = 1 \) are fundamental tools for simplifying expressions in problems involving trigonometric substitution.

Use the trigonometric substitution to write the algebraic expression as a trigonometric function of 𝜃, where 0 < 𝜃 < 𝜋/2: √(4 − x²), x = 2 cos(𝜃)

Learn how to use trigonometric substitution to express √(4 − x²) as a trigonometric function of 𝜃. This detailed step-by-step solution uses the substitution x = 2 cos(𝜃) and applies the Pythagorean identity to simplify the expression. Suitable for students in Grades 11-12.

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