In a right-angled triangle, we identify sides relative to a given acute angle:

• Hypotenuse: The longest side, opposite the right angle
• Opposite side: The side opposite to the angle we are considering
• Adjacent side: The side next to the angle we are considering (not the hypotenuse)

The Three Basic Ratios

For an acute angle A in a right-angled triangle:

$\sin A = \frac{\text{Opposite}}{\text{Hypotenuse}}$

$\cos A = \frac{\text{Adjacent}}{\text{Hypotenuse}}$

$\tan A = \frac{\text{Opposite}}{\text{Adjacent}}$

Memory Aid: SOH CAH TOA

Remember the mnemonic:

• SOH: Sine = Opposite ÷ Hypotenuse
• CAH: Cosine = Adjacent ÷ Hypotenuse
• TOA: Tangent = Opposite ÷ Adjacent

In right-angled triangle ABC, angle C = 90°, AB = 13 cm, BC = 5 cm, and AC = 12 cm.

Find: (a) sin A (b) cos A (c) tan A

Solution

First, identify the sides relative to angle A:

• Opposite side to angle A = BC = 5 cm
• Adjacent side to angle A = AC = 12 cm
• Hypotenuse = AB = 13 cm

(a) $\sin A = \frac{5}{13}$

(b) $\cos A = \frac{12}{13}$

(c) $\tan A = \frac{5}{12}$

Angle of Elevation

When you look upward from horizontal level to observe an object, the angle formed between the horizontal line and your line of sight is called the angle of elevation.

For example, if you stand on the ground and look up at the top of a tree, the angle between your horizontal view and the tree is the angle of elevation.

Angle of Depression

When you look downward from horizontal level to observe an object below you, the angle formed between the horizontal line and your line of sight is called the angle of depression.

For example, if you stand on a balcony and look down at a person on the ground, the angle between your horizontal view and the person is the angle of depression.

Important Property

The angle of elevation and angle of depression are equal when measured from horizontal lines that are parallel.

From the top of a tower, the angle of depression of a point on the ground 10 m from the base of the tower is 60°. How high is the tower?

Solution

Let A be the top of the tower, B be the base, and C be the point on the ground.

In triangle ABC:

• Angle of depression at A = 60° means angle C (the angle at ground level) also has tangent 60°
• Distance BC = 10 m

$\tan 60° = \frac{\text{height}}{\text{base}}$

$\tan 60° = \frac{AB}{10}$

$AB = 10 \times \tan 60°$

$AB = 10 \times 1.732 = 17.32 \text{ m}$

The tower is 17.32 m high.

1. Always draw a diagram when solving trigonometry problems—label the right angle, identify the angle you are working with, and mark the opposite, adjacent, and hypotenuse sides.

2. Check which ratio to use: If you know the opposite side and need the hypotenuse, use sine. If you know adjacent and need hypotenuse, use cosine. If you need the ratio of opposite to adjacent, use tangent.

3. For angle of elevation/depression problems: Draw the horizontal line from the observer's eye level and use the appropriate trigonometric ratio.

Trigonometry is used in construction work across Tanzania. For example, a builder in Dar es Salaam measuring the slope of a roof uses the angle of elevation to calculate the correct length of roofing materials needed. If the roof rises 3 meters vertically over a horizontal distance of 4 meters, the builder can use tan θ = 3/4 to find the roof angle, ensuring materials are cut at the correct angle for proper fitting.