Trigonometry

Core Concepts

Trigonometric Ratios

Trigonometry is the branch of mathematics that explores the relationships between the angles and sides of triangles. In a right-angled triangle, the sides are named relative to a specific acute angle, usually denoted by $\theta$:

• Hypotenuse: The longest side, directly opposite the $90^\circ$ right angle.
• Opposite: The side directly across from the angle $\theta$.
• Adjacent: The side next to the angle $\theta$, connecting it to the right angle.

The three primary trigonometric ratios define the relationships between these sides:

1. Sine ($\sin$): $\sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}}$
2. Cosine ($\cos$): $\cos \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}}$
3. Tangent ($\tan$): $\tan \theta = \frac{\text{Opposite}}{\text{Adjacent}}$

A common and highly effective mnemonic to remember these equations is SOH CAH TOA.

Trigonometric Ratios of Special Angles

Certain angles—specifically $30^\circ$, $45^\circ$, and $60^\circ$—appear frequently in geometry and trigonometry. Their ratios can be derived exactly using fundamental geometric shapes and are usually expressed in exact surd (radical) form.

• For $45^\circ$:

Consider an isosceles right-angled triangle with two equal sides of length $1$. By Pythagoras' theorem, the hypotenuse is $\sqrt{1^2 + 1^2} = \sqrt{2}$. $$ \sin 45^\circ = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2} $$ $$ \cos 45^\circ = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2} $$ $$ \tan 45^\circ = \frac{1}{1} = 1 $$

• For $30^\circ$ and $60^\circ$:

Consider an equilateral triangle of side $2$, bisected down the middle to form a right-angled triangle with a hypotenuse of $2$, a base of $1$, and a height of $\sqrt{2^2 - 1^2} = \sqrt{3}$. $$ \sin 30^\circ = \frac{1}{2}, \quad \cos 30^\circ = \frac{\sqrt{3}}{2}, \quad \tan 30^\circ = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3} $$ $$ \sin 60^\circ = \frac{\sqrt{3}}{2}, \quad \cos 60^\circ = \frac{1}{2}, \quad \tan 60^\circ = \frac{\sqrt{3}}{1} = \sqrt{3} $$

Trigonometric Tables

For angles other than special angles, mathematicians and students traditionally use Trigonometric Tables to look up approximate decimal values for sine, cosine, and tangent up to four decimal places. When using tables:

• Read down the leftmost column to find the degree of the angle.
• Read across the top row to find the fractional part (often in minutes or tenths of a degree).
• Find the intersection of the row and column to extract the ratio.
• Inverse usage: Tables can be read "backwards" to find an unknown angle if its ratio is given (e.g., finding $\theta$ given $\sin \theta = 0.5312$).

Angles of Elevation and Depression

Trigonometry is highly useful for measuring distances vertically and horizontally in the real world.

• Angle of Elevation: The upward angle measured from the horizontal line of sight of an observer to an object situated above them.
• Angle of Depression: The downward angle measured from the horizontal line of sight of an observer to an object situated below them.

A critical geometric principle is that horizontal lines of sight are parallel. Because of alternate interior angles, the angle of elevation from point $A$ to point $B$ is mathematically equivalent to the angle of depression from point $B$ to point $A$.

Worked Examples

Example 1: Finding an Unknown Side Using Special Angles A ladder leans against a vertical wall. The ladder is $10\text{ m}$ long and makes an angle of $60^\circ$ with the horizontal ground. How high up the wall does the ladder reach?

Solution: Let $h$ be the height of the wall. The ladder forms the hypotenuse: $10\text{ m}$. The height of the wall is the side opposite to the $60^\circ$ angle. Using SOH: $$ \sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}} $$ $$ \sin 60^\circ = \frac{h}{10} $$ Substitute the exact value for $\sin 60^\circ$: $$ \frac{\sqrt{3}}{2} = \frac{h}{10} $$ $$ h = 10 \times \frac{\sqrt{3}}{2} $$ $$ h = 5\sqrt{3}\text{ m} $$ The ladder reaches $5\sqrt{3}\text{ m}$ up the wall.

Example 2: Angle of Depression A bird perched on top of a tree that is $15\text{ m}$ tall spots a worm on the ground at an angle of depression of $45^\circ$. How far is the worm from the base of the tree?

Solution: Let $d$ be the horizontal distance from the worm to the tree. The angle of depression from the bird is $45^\circ$. Therefore, the angle of elevation from the worm to the bird is also $45^\circ$. The tree's height represents the opposite side ($15\text{ m}$), and the horizontal distance is the adjacent side. Using TOA: $$ \tan \theta = \frac{\text{Opposite}}{\text{Adjacent}} $$ $$ \tan 45^\circ = \frac{15}{d} $$ Substitute the exact value for $\tan 45^\circ$: $$ 1 = \frac{15}{d} $$ $$ d = 15\text{ m} $$ The worm is $15\text{ m}$ from the base of the tree.

NECTA Exam Focus

In the NECTA CSEE Basic Mathematics exams, Trigonometry questions typically appear in Section A or B, strongly focusing on the real-world application of the SOH CAH TOA rules.

• Recurring Themes: A classic exam setup involves describing a vertical structure (such as a tower, building, or tree) and an object on the horizontal ground. The problem will usually provide an angle of elevation or depression alongside one known measurement, asking the candidate to solve for the missing horizontal distance or vertical height.
• Exact Surd Values: A recurring instruction in NECTA papers is "leave the answer in surd form." This tests whether students have memorized the exact trigonometric ratios for special angles ($30^\circ$, $45^\circ$, and $60^\circ$). Relying strictly on a calculator to produce decimal answers will lose marks when this instruction is present.
• Common Pitfalls:
• Misplacing the Angle of Depression: The most common mistake students make is drawing the angle of depression relative to the vertical object (like a wall or tower) instead of relative to the horizontal line of sight.
• Ignoring Final Output Constraints: Providing a decimal approximation (like $1.732$) instead of keeping $\sqrt{3}$ when instructed to use surd form.

Practice Problems

1. From the top of a tower which is 50 m high, the angle of depression of a car parked on the ground is $30^{\circ}$. How far is the car from the base of the tower? Leave the answer in surd form.

2. Misumbwi, Shuma and Kiyando contributed 770,000, 560,000 and 1,050,000 shillings respectively to start a business. Find the ratio of their contribution in its simplest form. (Note: Review of standard ratios which share the foundational arithmetic of proportional representation used in Trigonometry).

Subtopics

• Trigonometric functions
• Graphs of sine and cosine
• Sine rule
• Cosine rule
• Applications

Crosswalk Notes

Cross-version relationships are drafted in data/curricula/crosswalks/csee-basic-mathematics-2005-to-mathematics-2023.json. Partial and 2005-only mappings remain reviewable.