Trigonometry is a powerful tool for finding unknown angles and side lengths in right-angled triangles. This topic is frequently combined with Pythagoras' Theorem and appears in questions about elevation, depression, and bearings.

1. The Three Basic Ratios: SOH CAH TOA

This acronym is the most important thing to remember. For any acute angle θ in a right-angled triangle:

• Hypotenuse (H): The longest side, opposite the right angle.

• Opposite (O): The side directly opposite the angle θ.

• Adjacent (A): The side next to the angle θ (that is not the hypotenuse).

The ratios are:

• SOH: Sin θ = Opposite / Hypotenuse

• CAH: Cos θ = Adjacent / Hypotenuse

• TOA: Tan θ = Opposite / Adjacent

2. Special Angles: 30°, 45°, 60°

You should memorize the exact values for these common angles. Exam questions using these angles often expect you to leave the answer in surd form (with √ signs).

Exam Tip: If you see these angles in a question, you likely won't need the main trigonometric tables. Use these exact values for a precise answer.

3. Angles of Elevation and Depression

These are key concepts for word problems. Both are measured from the horizontal line of sight.

• Angle of Elevation: The angle you look UP to see an object.

• Angle of Depression: The angle you look DOWN to see an object.

Crucial Trick: The angle of depression from point A to point B is equal to the angle of elevation from point B to point A. They form alternate angles ('Z' shape) with parallel horizontal lines. This is used in almost every exam problem.

4. Bearings

Bearings are used to describe direction in a horizontal plane. Remember the three rules:

1. Measure from the North line.

2. Measure in a clockwise direction.

3. Always write the angle using three digits (e.g., 40° is written as 040°).

5. The 5-Step Problem-Solving Strategy

For any word problem involving trigonometry:

1. Draw a Diagram: Sketch the situation. Mark all right angles, lengths, and angles you are given.

2. Identify the Triangle: Choose the right-angled triangle that contains the side or angle you need to find.

3. Label the Sides: Based on the angle you are using, label the sides as Opposite (O), Adjacent (A), and Hypotenuse (H).

4. Choose the Ratio: Use SOH CAH TOA to select the correct ratio that connects the side you know with the side you want to find.

5. Set up and Solve: Write the equation and solve for the unknown. You may need to use the trigonometric tables or your knowledge of special angles.

Common Pitfalls & Exam Tips

• Pitfall 1: Mixing up the sides. Always label O, A, and H relative to the angle you are using. If you switch to the other acute angle in the triangle, the Opposite and Adjacent sides will swap.

• Pitfall 2: Observer's Height. If a problem mentions the height of a person (e.g., "the angle of elevation from the eyes of a person 1.5 m tall"), you must include this height in a rectangle at the bottom of your diagram and add it to your final answer for the total height.

• Exam Tip 1: Two-Part Problems. Often, you'll need to solve for a side in one triangle first, and then use that side's length in a second triangle to get the final answer.

• Exam Tip 2: Logarithms. In complex calculations (e.g., x = 10 / sin(57° 32')), the exam may require you to use logarithm tables to find the final value. Be prepared to switch between trigonometry and logarithms.