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### **Question 1:**
Find the value of \(\sin x\), \(\cos x\), and \(\tan x\).

Given a right triangle with the following sides:
- Opposite side (to angle \(x\)) = 3 cm
- Hypotenuse = 5 cm

We can use the following trigonometric definitions:
- \(\sin x = \frac{\text{opposite}}{\text{hypotenuse}}\)
- \(\cos x = \frac{\text{adjacent}}{\text{hypotenuse}}\)
- \(\tan x = \frac{\text{opposite}}{\text{adjacent}}\)

First, calculate the adjacent side using the Pythagorean theorem:
\[
\text{adjacent}^2 + 3^2 = 5^2
\]
\[
\text{adjacent}^2 + 9 = 25
\]
\[
\text{adjacent}^2 = 16 \quad \Rightarrow \quad \text{adjacent} = 4 \text{ cm}
\]

Now, calculate the trigonometric ratios:
- \(\sin x = \frac{3}{5} = 0.6\)
- \(\cos x = \frac{4}{5} = 0.8\)
- \(\tan x = \frac{3}{4} = 0.75\)

### **Question 2:**
Calculate the angle between \(\overrightarrow{OP}\) and \(\overrightarrow{OQ}\) for the following equations:
- \(P: 2x - y = 3\)
- \(Q: 3x + 2y = 0\)

The angle between two lines can be found using the formula:
\[
\tan \theta = \left| \frac{m_1 - m_2}{1 + m_1m_2} \right|
\]
where \(m_1\) and \(m_2\) are the slopes of the lines.

1. Find the slope (\(m_1\)) of line \(P: 2x - y = 3\):
   \[
   y = 2x - 3 \quad \Rightarrow \quad m_1 = 2
   \]

2. Find the slope (\(m_2\)) of line \(Q: 3x + 2y = 0\):
   \[
   2y = -3x \quad \Rightarrow \quad y = -\frac{3}{2}x \quad \Rightarrow \quad m_2 = -\frac{3}{2}
   \]

Substitute \(m_1\) and \(m_2\) into the formula:
\[
\tan \theta = \left| \frac{2 - \left(-\frac{3}{2}\right)}{1 + 2\left(-\frac{3}{2}\right)} \right| = \left| \frac{2 + \frac{3}{2}}{1 - 3} \right| = \left| \frac{\frac{4}{2} + \frac{3}{2}}{-2} \right| = \left| \frac{\frac{7}{2}}{-2} \right| = \frac{7}{4}
\]
Thus, \(\theta = \tan^{-1}\left(\frac{7}{4}\right)\).

### **Question 3:**
Find the angles of the triangle with \(a = 8.7 \, \text{cm}\), \(b = 10.7 \, \text{cm}\), and \(c = 11.8 \, \text{cm}\).

This is a non-right triangle, so we can use the **Cosine Rule** to find the angles. The Cosine Rule states:
\[
\cos A = \frac{b^2 + c^2 - a^2}{2bc}, \quad \cos B = \frac{a^2 + c^2 - b^2}{2ac}, \quad \cos C = \frac{a^2 + b^2 - c^2}{2ab}
\]

Let's calculate one angle as an example:
1. **Angle \(A\)**:
   \[
   \cos A = \frac{10.7^2 + 11.8^2 - 8.7^2}{2 \times 10.7 \times 11.8}
   \]
   \[
   = \frac{114.49 + 139.24 - 75.69}{2 \times 10.7 \times 11.8} 
   = \frac{178.04}{252.28} \approx 0.7058
   \]
   \[
   A \approx \cos^{-1}(0.7058) \approx 45.0^\circ
   \]

Repeat the process for angles \(B\) and \(C\) using the same method.

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Would you like more details or have any questions?

Here are some related questions:
1. How can the tangent of an angle help in determining the slope between two lines?
2. What would be the value of \(\sin x\) if the triangle was isosceles with both legs equal?
3. How would you find the area of the triangle in Question 3 using trigonometric methods?
4. Could the sine rule be used to solve Question 3 instead? How?
5. What is the relationship between the angles and sides in a right triangle?

**Tip:** Always double-check your triangle's classification (right, isosceles, or scalene) before choosing a method to solve for angles or sides.

Learn how to find trigonometric ratios and angles in a right triangle and calculate the angle between two lines using trigonometric formulas. Detailed step-by-step solutions are provided for high school students.

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