What is Trigonometry?
A subfield of mathematics called trigonometry examines the connections between triangles’ angles and sides. It is widely used in various fields such as physics, engineering, architecture, and even navigation. In GCSE and IGCSE mathematics, trigonometry is a crucial topic that helps students understand how to solve problems involving right-angled and non-right-angled triangles.
Basic Trigonometric Ratios
For right-angled triangles, the three primary trigonometric ratios are:
- Sine (sin) = Opposite / Hypotenuse
- Cosine (cos) = Adjacent / Hypotenuse
- Tangent (tan) = Opposite / Adjacent
These ratios help calculate missing angles and side lengths of right-angled triangles.
Trigonometry Tricks and Tips for GCSE and IGCSE
- Remember SOHCAHTOA
The acronym SOHCAHTOA helps students recall the three trigonometric ratios:
- SOH: Sin = Opposite / Hypotenuse
- CAH: Cos = Adjacent / Hypotenuse
- TOA: Tan = Opposite / Adjacent
Always use this to determine which ratio to apply in a given problem.
- Use the Inverse Trigonometric Functions for Angles
If you are given two sides and need to find an angle, use the inverse trigonometric functions:
- sin⁻¹, cos⁻¹, and tan⁻¹ on a calculator.
- Example: If sin (θ) = 0.5, then θ = sin⁻¹ (0.5) = 30°.
- Pythagoras’ Theorem for Missing Sides
For right-angled triangles: a² + b² = c² (where c is the hypotenuse)
- Use this theorem when two sides are given, and the third side needs to be found.
- Use the Sine and Cosine Rules for Non-Right-Angled Triangles
When dealing with non-right-angled triangles, these formulas are essential:
Sine Rule:
(a/sin A) = (b/sin B) = (c/sin C)
- Used to indicate that you know two sides and a non-included angle (SSA) or two angles and one side (AAS or ASA).
Cosine Rule:
a^2 + b^2 – 2ab × cos (c) = c^2
- Used when you know two sides and an included angle (SAS) or all three sides (SSS).
- Memorize Key Trigonometric Values
Some commonly used values:
- Cos 30° = 0.866, tan 30° = 0.577, and sin 30° = 0.5
- Tan 45° = 1, cos 45° = 0.707, and sin 45° = 0.707
- Cos 60° = 0.5, tan 60° = 1.732, and sin 60° = 0.866
Recalling patterns such as the square root approach for sine and cosine values is a helpful tip:
30° is equal to √1/2, 45° to √2/2, and 60° to √3/2.