https://file.aiquickdraw.com/mathbot/file/172729160889209t2nasd.jpg

To solve this, we are given:

- \( \theta \) is in the third quadrant.
- \( \tan \theta = \frac{5}{8} \).
- We are to find the exact value of \( \sin \frac{\theta}{2} \).

### Step-by-step process:

#### 1. **Use the identity for half-angle of sine**:
The half-angle identity for sine is:

\[
\sin \frac{\theta}{2} = \pm \sqrt{\frac{1 - \cos \theta}{2}}
\]

Since \( \theta \) is in the third quadrant, \( \frac{\theta}{2} \) will be in the second quadrant (for angles between \( 180^\circ \) and \( 270^\circ \), their halves fall between \( 90^\circ \) and \( 135^\circ \)). In the second quadrant, sine is positive. Therefore, we'll use the positive sign for the square root.

#### 2. **Find \( \cos \theta \) using \( \tan \theta \)**:
We can find \( \cos \theta \) by first determining the hypotenuse in a right triangle using the tangent ratio. Since:

\[
\tan \theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{5}{8}
\]

Using the Pythagorean theorem, the hypotenuse \( r \) is:

\[
r = \sqrt{5^2 + 8^2} = \sqrt{25 + 64} = \sqrt{89}
\]

Thus, the cosine of \( \theta \) is:

\[
\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{8}{\sqrt{89}}
\]

Since \( \theta \) is in the third quadrant, where cosine is negative, we have:

\[
\cos \theta = -\frac{8}{\sqrt{89}}
\]

#### 3. **Substitute \( \cos \theta \) into the half-angle identity**:
Now substitute \( \cos \theta = -\frac{8}{\sqrt{89}} \) into the half-angle formula:

\[
\sin \frac{\theta}{2} = \sqrt{\frac{1 - (-\frac{8}{\sqrt{89}})}{2}} = \sqrt{\frac{1 + \frac{8}{\sqrt{89}}}{2}}
\]

#### 4. **Simplify the expression**:
Combine the terms inside the square root:

\[
\sin \frac{\theta}{2} = \sqrt{\frac{\frac{\sqrt{89}}{\sqrt{89}} + \frac{8}{\sqrt{89}}}{2}} = \sqrt{\frac{\sqrt{89} + 8}{2\sqrt{89}}}
\]

#### 5. **Rationalize the denominator** (optional for further simplification):
You can match this with the options provided, but without fully rationalizing, it seems that the correct option would involve the terms we've calculated.

Do you need any further explanation or the specific choice marked from the options?

### Five follow-up questions:
1. What is the half-angle identity for cosine?
2. How do we determine the sign of trigonometric functions in different quadrants?
3. Can you derive the identity for tangent of half an angle?
4. How do we simplify expressions involving square roots?
5. Why is it important to rationalize the denominator in trigonometric problems?

### Tip:
Always check the quadrant of the angle when using trigonometric identities to determine the correct sign!

Consider the angle θ in the third quadrant. Given that 0° < θ < 360° and tan(θ) = 5/8, find the exact value of sin(θ/2).

This problem involves finding the exact value of sin(θ/2) given tan(θ) = 5/8 with θ in the third quadrant. It uses trigonometric identities such as the half-angle formula and the Pythagorean Theorem. Suitable for high school students studying advanced trigonometry.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation