Determine trigonometric ratios of angles and special angles

Trigonometric Ratios of Angles and Special Angles

Introduction

Trigonometry deals with the relationship between the sides and angles of right-angled triangles. In this topic, you will learn how to determine the three basic trigonometric ratios—sine, cosine, and tangent—for any angle, and also for special angles whose values are commonly used.

Basic Trigonometric Ratios

In a right-angled triangle, for a given acute angle (let's call it θ), the sides are named as follows:

• Hypotenuse: The longest side, opposite the right angle.
• Opposite side: The side opposite to angle θ.
• Adjacent side: The side next to angle θ (excluding the hypotenuse).

The three basic trigonometric ratios are defined as:

$\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{O}{H}$

$\cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{A}{H}$

$\tan\theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{O}{A}$

A useful mnemonic to remember these definitions is SOH CAH TOA:

• Sine = Opposite ÷ Hypotenuse
• Cosine = Adjacent ÷ Hypotenuse
• Tangent = Opposite ÷ Adjacent

Special Angles

Special angles are angles whose trigonometric ratios can be expressed in exact values using simple radicals. The special angles are 0°, 30°, 45°, 60°, and 90°.

These ratios are derived from two key geometric figures:

1. For 30° and 60°: Use an equilateral triangle with side length 2 units. The altitude creates a 30-60-90 right triangle.

2. For 45°: Use an isosceles right triangle with equal legs of 1 unit each.

Table of Trigonometric Ratios for Special Angles

Angle (θ)	sin θ	cos θ	tan θ
0°	0	1	0
30°	$\frac{1}{2}$	$\frac{\sqrt{3}}{2}$	$\frac{\sqrt{3}}{3}$
45°	$\frac{\sqrt{2}}{2}$	$\frac{\sqrt{2}}{2}$	1
60°	$\frac{\sqrt{3}}{2}$	$\frac{1}{2}$	$\sqrt{3}$
90°	1	0	undefined

Note: At 90°, the adjacent side becomes zero, so tangent is undefined (division by zero).

Worked Examples

Example 1: In a right-angled triangle ABC, ∠B = 90°, AB = 3 cm, BC = 4 cm, and AC = 5 cm. Find the value of sin A, cos A, and tan A.

Solution

First, identify the sides relative to angle A:

• The side opposite to angle A is BC = 4 cm
• The side adjacent to angle A is AB = 3 cm
• The hypotenuse is AC = 5 cm

Now apply the formulas:

$\sin A = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{BC}{AC} = \frac{4}{5}$

$\cos A = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{AB}{AC} = \frac{3}{5}$

$\tan A = \frac{\text{opposite}}{\text{adjacent}} = \frac{BC}{AB} = \frac{4}{3}$

Example 2: Find the exact value of $2\sin 60° + \cos 30°$.

Solution

From the table of special angles:

• $\sin 60° = \frac{\sqrt{3}}{2}$
• $\cos 30° = \frac{\sqrt{3}}{2}$

Therefore:

$2\sin 60° + \cos 30° = 2 \times \frac{\sqrt{3}}{2} + \frac{\sqrt{3}}{2}$

$= \sqrt{3} + \frac{\sqrt{3}}{2} = \frac{3\sqrt{3}}{2}$

Finding One Ratio When Another Is Given

Sometimes you will be given one trigonometric ratio and asked to find the others. Use the Pythagorean theorem:

$\sin^2\theta + \cos^2\theta = 1$

Example 3: Given that $\tan x = \frac{5}{12}$, find $\cos x$.

Solution

Since $\tan x = \frac{\text{opposite}}{\text{adjacent}} = \frac{5}{12}$, we can consider a right triangle with:

• Opposite side = 5 units
• Adjacent side = 12 units

Using Pythagorean theorem:

$\text{Hypotenuse} = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13$

Now:

$\cos x = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{12}{13}$

Real-life application

Trigonometric ratios are used in construction and surveying in Tanzania. For example, when building a roof or a ramp, builders use the tangent ratio to calculate the correct angle of inclination. A mason in Dar es Salaam determining the slope of a wheelchair ramp 3 meters long that rises 0.6 meters can use $\tan \theta = \frac{0.6}{3}$ to find the angle, ensuring the ramp meets accessibility standards.