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This worksheet focuses on trigonometric functions—particularly sine, cosine, and tangent—applied to right-angled triangles. To solve these problems, we can use the definitions of sine, cosine, and tangent based on the sides of a right triangle:

- **Sine (sin) θ = Opposite / Hypotenuse**
- **Cosine (cos) θ = Adjacent / Hypotenuse**
- **Tangent (tan) θ = Opposite / Adjacent**

I'll help you solve the problems from the worksheet.

Let's solve the first few examples as a guide:

### 1. 
\[
\sin x^\circ = \frac{4}{5}, \quad BC = ?
\]

Here, sine is given as \(\sin x^\circ = \frac{4}{5}\), which means:

\[
\frac{Opposite}{Hypotenuse} = \frac{4}{5}
\]

So, \(BC = 5\), and \(AB = 4\). To find \(AC\), we use the Pythagorean theorem:

\[
AC^2 = AB^2 + BC^2
\]
\[
AC^2 = 4^2 + 5^2 = 16 + 25 = 41
\]
\[
AC = \sqrt{41}
\]

### 2. 
\[
\sin x^\circ = \frac{5}{13}, \quad AB = ?
\]

The sine function gives us:

\[
\frac{Opposite}{Hypotenuse} = \frac{5}{13}
\]

So, \(AB = 5\), and \(AD = 13\). To find \(AC\), we again use the Pythagorean theorem:

\[
AC^2 = AB^2 + BC^2 = 5^2 + 13^2 = 25 + 169 = 194
\]
\[
AC = \sqrt{194}
\]

Would you like me to proceed with the rest or explain in more detail for a specific problem? Let me know!

Here are some follow-up questions to expand on this:
1. What is the relation between sine, cosine, and tangent in terms of their respective sides?
2. How does the Pythagorean theorem assist in solving trigonometry problems involving right triangles?
3. What are the common values for sine, cosine, and tangent for specific angles like 30°, 45°, and 60°?
4. Can you derive cosine and tangent from sine values?
5. How do we apply inverse trigonometric functions to calculate angles from side lengths?

**Tip:** Always remember to check if your triangle satisfies the Pythagorean theorem when solving for unknown sides.

Solve the trigonometric problems on right-angled triangles from the worksheet.

Solve trigonometric problems involving sine, cosine, and tangent functions for right-angled triangles. This worksheet includes calculations using sin, cos, and tan along with the Pythagorean theorem to find missing side lengths. Suitable for students in Grades 9-11.

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