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Please provide step-by-step solutions for  all the questions   in Task 5: Trigonometry

Here are the step-by-step solutions for the trigonometry questions in **Task 5**:

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### **Question 1**

**Problem:** The diagram shows two right-angled triangles ABC and APQ. Using triangle ABC, write down the value of \(\tan x\).

**Solution:**

In a right-angled triangle, the tangent of an angle is given by:

\[
\tan x = \frac{\text{Opposite}}{\text{Adjacent}}
\]

Here, assuming we have the lengths of the sides opposite and adjacent to angle \(x\):

\[
\tan x = \frac{\text{Length of side opposite to } x}{\text{Length of side adjacent to } x}
\]

Since the specific side lengths aren’t provided in the question text, you would use the ratio of these sides from the triangle provided in the diagram.

---

### **Question 2**

**Problem:** ABC is a right-angled triangle. D is a point on AC, and BD is perpendicular to AC. It can be shown that \(\cos \theta = \frac{x}{16}\). Work out the value of \(x\).

**Solution:**

Given:
\[
\cos \theta = \frac{x}{16}
\]

In a right-angled triangle, \(\cos \theta = \frac{\text{Adjacent side}}{\text{Hypotenuse}}\).

Thus, if we know that \(\cos \theta = \frac{x}{16}\), \(x\) must represent the length of the adjacent side, and the hypotenuse is 16 cm.

Therefore:

\[
x = 16 \cos \theta
\]

---

### **Question 3**

**Problem:** PQRS is a trapezium. Work out the perimeter of PQRS. (Answer to 1 decimal place).

**Solution:**

For a trapezium, the perimeter is the sum of all sides:

\[
\text{Perimeter} = PQ + QR + RS + SP
\]

Without specific side lengths given in the problem, you would calculate each length, potentially using trigonometry, if the lengths are not directly provided, and then sum them up.

If side lengths are provided or can be derived from the diagram, substitute the values:

\[
\text{Perimeter} = \text{sum of all sides}
\]

---

### **Question 4**

**Problem:** A triangle has sides 10.2 cm, 6.8 cm, and 5.7 cm. Work out the area of the triangle.

**Solution:**

Use **Heron's formula** for the area of a triangle, where \(s\) is the semi-perimeter:

1. Compute the semi-perimeter \(s\):

\[
s = \frac{10.2 + 6.8 + 5.7}{2} = 11.35 \, \text{cm}
\]

2. Use Heron's formula:

\[
\text{Area} = \sqrt{s(s-a)(s-b)(s-c)}
\]

Substitute \(a = 10.2\), \(b = 6.8\), and \(c = 5.7\):

\[
\text{Area} = \sqrt{11.35(11.35-10.2)(11.35-6.8)(11.35-5.7)}
\]

Calculate:

\[
\text{Area} \approx \sqrt{11.35 \times 1.15 \times 4.55 \times 5.65} \approx 17.3 \, \text{cm}^2
\]

---

### **Question 5**

**Problem:** Use the cosine rule to express \(p\) in terms of \(m\).

**Solution:**

The **cosine rule** states:

\[
c^2 = a^2 + b^2 - 2ab\cos C
\]

Assuming \(p\) is the side opposite angle \(C\) and sides \(a\) and \(b\) are related to \(m\):

\[
p^2 = a^2 + b^2 - 2ab\cos C
\]

If \(a = b = m\), then:

\[
p^2 = m^2 + m^2 - 2m^2\cos C
\]

\[
p = \sqrt{2m^2(1-\cos C)}
\]

Or more generally, the expression for \(p\) depends on the specific triangle’s sides and angle \(C\) given in the problem.

---

Would you like more details, or do you have any questions?

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### Relative Questions:
1. What is the sine rule and when is it used in trigonometry?
2. How do you find the height of a triangle when only sides are known?
3. What is Heron's formula and how is it derived?
4. How do you determine the angle between two planes in three-dimensional geometry?
5. How can the cosine rule be applied to non-right triangles?

### Tip:
**Tip:** When working with trigonometric identities, always double-check the angle type (degrees vs. radians) to avoid calculation errors.

Explore detailed step-by-step solutions for trigonometry questions in Task 5, covering topics such as tangent formulas, Heron's formula for area calculation, and the cosine rule. Suitable for high school students studying trigonometry.

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