If cos theta = 4/5 find Sin 20 and Cos30.

To solve this problem, we need to address the components separately.

### 1. Finding \( \sin \theta \):
Given that \( \cos \theta = \frac{4}{5} \), we can use the Pythagorean identity:

\[
\sin^2 \theta + \cos^2 \theta = 1
\]

Substituting the given value:

\[
\sin^2 \theta + \left(\frac{4}{5}\right)^2 = 1
\]

\[
\sin^2 \theta + \frac{16}{25} = 1
\]

\[
\sin^2 \theta = 1 - \frac{16}{25} = \frac{25}{25} - \frac{16}{25} = \frac{9}{25}
\]

Taking the square root of both sides:

\[
\sin \theta = \frac{3}{5}
\]

### 2. Finding \( \sin 20^\circ \) and \( \cos 30^\circ \):
These are standard trigonometric values:

\[
\sin 20^\circ \approx 0.3420
\]

\[
\cos 30^\circ = \frac{\sqrt{3}}{2} \approx 0.8660
\]

So:

- \( \sin 20^\circ \approx 0.3420 \)
- \( \cos 30^\circ \approx 0.8660 \)

Would you like any additional explanations or further questions on these topics?

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Here are five related questions:
1. How do you derive the exact value of \( \sin 30^\circ \) and \( \cos 60^\circ \)?
2. What is the Pythagorean identity and how is it applied in trigonometry?
3. How do you calculate trigonometric functions for non-standard angles?
4. What are the exact values of sine and cosine for common angles like 45°, 60°, and 90°?
5. How can the unit circle be used to find trigonometric functions of any angle?

**Tip:** For non-standard angles, trigonometric tables or a calculator can provide approximate values, but understanding standard angles helps in many problems.

Learn how to solve trigonometric equations involving the cosine of 30 degrees and sine of 20 degrees. Understand the Pythagorean identity and apply it to find exact values. Suitable for high school students.

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