Trigonometry

Introduction to Trigonometry & Bearings

Trigonometry is a fundamental branch of mathematics that studies the relationships between the sides and angles of triangles. Its applications are vast, from engineering and architecture to astronomy and navigation. In Cambridge O Level Mathematics, we focus on the practical application of these principles to solve problems in two dimensions.

1. Right-Angled Triangles: SOH CAH TOA

The foundation of trigonometry lies in the right-angled triangle. We label the sides in relation to a specific angle, θ (theta):

* Hypotenuse (H): The longest side, always 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 three basic trigonometric ratios are defined as follows, easily remembered by the mnemonic SOH CAH TOA:

* SOH: Sin(θ) = Opposite / Hypotenuse

* CAH: Cos(θ) = Adjacent / Hypotenuse

* TOA: Tan(θ) = Opposite / Adjacent

Process: Finding an Unknown Side

1. Identify the known angle and known side.
2. Label the sides (O, A, H) relative to the known angle.
3. Choose the correct ratio (sin, cos, or tan) that connects the knowns with the unknown side.
4. Substitute the values and solve the equation.

*Example:* In a right-angled triangle, if an angle is 40° and the adjacent side is 12 cm, find the opposite side (x).

We have the Adjacent (A) and need the Opposite (O). This uses TOA.

`tan(40°) = Opposite / Adjacent`

`tan(40°) = x / 12`

`x = 12 × tan(40°) ≈ 10.1 cm`

Process: Finding an Unknown Angle

1. Identify the two known sides.
2. Label them relative to the angle you want to find.
3. Choose the correct ratio.
4. Use the inverse trigonometric function (sin⁻¹, cos⁻¹, tan⁻¹) to find the angle.

*Example:* If the opposite side is 5 cm and the hypotenuse is 8 cm, find the angle θ.

We have Opposite (O) and Hypotenuse (H). This uses SOH.

`sin(θ) = 5 / 8`

`θ = sin⁻¹(5 / 8) ≈ 38.7°`

Common Exam Trap: Always ensure your calculator is in **Degree (DEG) mode**, not Radian (RAD) or Gradient (GRAD) mode, as O Level questions are exclusively in degrees.

2. Non-Right-Angled Triangles: Sine & Cosine Rules

When a triangle does not have a 90° angle, SOH CAH TOA cannot be used directly. We use the Sine and Cosine rules instead. The convention is to label angles with capital letters (A, B, C) and their opposite sides with corresponding lowercase letters (a, b, c).

The Sine Rule

This rule is used when you know a side and its opposite angle (a 'matching pair').

Formula:

`a/sin A = b/sin B = c/sin C`

When to use it:

* Given two angles and any side (AAS).

* Given two sides and a non-included angle (SSA).

*Example (finding a side):* In triangle ABC, A = 48°, B = 57°, and side c = 15 cm. Find side a.

First, find angle C: C = 180° - 48° - 57° = 75°.

Now use the Sine Rule: `a/sin A = c/sin C`

`a / sin(48°) = 15 / sin(75°)`

`a = (15 × sin(48°)) / sin(75°) ≈ 11.5 cm`

The Cosine Rule

This rule connects all three sides with one angle.

Formula (to find a side):

`a² = b² + c² - 2bc cos A`

Formula (rearranged to find an angle):

`cos A = (b² + c² - a²) / 2bc`

When to use it:

* Given two sides and the included angle (SAS) to find the third side.

* Given all three sides (SSS) to find any angle.

*Example (finding an angle):* A triangular plot of land in a village near Faisalabad has sides of 50m, 70m, and 80m. Find the largest angle.

The largest angle is opposite the longest side (80m). Let's call it angle A.

`cos A = (50² + 70² - 80²) / (2 × 50 × 70)`

`cos A = (2500 + 4900 - 6400) / 7000 = 1000 / 7000`

`A = cos⁻¹(1000 / 7000) ≈ 81.8°`

3. Applications: Bearings and Angles of Elevation/Depression

Bearings

A bearing is an angle used in navigation to describe direction. It follows three strict rules:

1. It is measured from the North line.
2. It is measured in a clockwise direction.
3. It is always written as a 3-figure number (e.g., 075° instead of 75°).

*Example:* The bearing of Lahore from Karachi is approximately 035°. This means if you are in Karachi, you face North and turn 35° clockwise to face Lahore.

A key skill is finding the 'back bearing'. If the bearing of B from A is θ, the bearing of A from B is found by using properties of parallel North lines. The rule is:

* If θ < 180°, the back bearing is θ + 180°.

* If θ > 180°, the back bearing is θ - 180°.

Angles of Elevation and Depression

These are angles measured relative to the horizontal.

* Angle of Elevation: The angle measured upwards from a horizontal line to an object above.

* *Practical Use:* An observer on the ground at the Badshahi Mosque in Lahore looks up at the top of a minaret. The angle their line of sight makes with the ground is the angle of elevation.

* Angle of Depression: The angle measured downwards from a horizontal line to an object below.

* *Practical Use:* Someone at the top of Minar-e-Pakistan looks down at a visitor on the ground. The angle their line of sight makes with the horizontal is the angle of depression.

Common Misconception: The angle of depression is outside the main triangle of the problem. However, because the horizontal line is parallel to the ground, the angle of depression from point A to point B is equal to the angle of elevation from point B to point A (they are alternate angles).

Solving Strategy: For all application problems, the first and most crucial step is to **draw a clear, large, and well-labelled diagram**. This will help you identify the triangles and choose the correct trigonometric rule to solve the problem.