Defining the Trigonometric Ratios (Grade 10 NSC Matric Mathematics): Revision Notes

Worked Example: Triangle Side Identification

Let's work through a complete example to demonstrate how to identify triangle sides and set up trigonometric ratios.

Question: For the right-angled triangle shown, identify the hypotenuse, opposite, and adjacent sides with respect to angle θ, then set up the trigonometric ratios.

Step 1: Identify the triangle components

First, locate the right angle - this is always marked with a small square symbol. The hypotenuse is always the side directly opposite this right angle and is the longest side of the triangle.

Next, focus on angle θ. The opposite side is the side directly across from angle θ (not touching the angle). The adjacent side is the side that is next to angle θ and helps form this angle (but is not the hypotenuse).

For angle θ:

• Hypotenuse: AC (longest side, opposite the right angle)
• Opposite: BC (the side across from angle θ)
• Adjacent: AB (the side next to angle θ)

Step 2: Set up the trigonometric ratios

Now we can write the three trigonometric ratios for angle θ:

$\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{BC}{AC}$

$\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{AB}{AC}$

$\tan \theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{BC}{AB}$

Step 3: Consider the other angle

If we were asked about angle α (the other non-right angle), the opposite and adjacent sides would switch positions, but the hypotenuse remains the same:

For angle α:

• Hypotenuse: AC (still the same)
• Opposite: AB (now opposite to angle α)
• Adjacent: BC (now adjacent to angle α)

The trigonometric ratios for angle α would be:

$\sin \alpha = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{AB}{AC}$

$\cos \alpha = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{BC}{AC}$

$\tan \alpha = \frac{\text{opposite}}{\text{adjacent}} = \frac{AB}{BC}$

Worked Example: Calculating Trigonometric Ratios

Question: In triangle ABC with right angle at B, side AB = 3 units, side BC = 4 units, and side AC = 5 units. Calculate sin A, cos A, and tan A.

Step 1: Identify the sides relative to angle A

• Hypotenuse: AC = 5 units (opposite the right angle)
• Opposite (to angle A): BC = 4 units
• Adjacent (to angle A): AB = 3 units

Step 2: Apply the trigonometric ratios

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

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

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

Worked Example: Finding Unknown Sides

Question: In a right-angled triangle, angle θ = 30°, the hypotenuse = 10 cm, and you need to find the opposite side.

Step 1: Identify what you know and what you need

• Known: θ = 30°, hypotenuse = 10 cm
• Unknown: opposite side
• Required ratio: sin θ (involves opposite and hypotenuse)

Step 2: Set up the equation

$\sin 30° = \frac{\text{opposite}}{\text{hypotenuse}}$

$\sin 30° = \frac{\text{opposite}}{10}$

Step 3: Solve for the unknown

$\text{opposite} = 10 \times \sin 30°$

$\text{opposite} = 10 \times 0.5 = 5 \text{ cm}$