Trigonometric Ratios and Right-Angled Triangles

The Three Fundamental Trigonometric Ratios

Once you have identified the sides, you can use trigonometric ratios to link the angle to the side lengths. There are three primary ratios: Sine, Cosine, and Tangent.

1. The Sine Formula

The Sine of an angle (written as $\sin x$) represents the ratio of the length of the opposite side to the length of the hypotenuse.

$\sin x = \frac{\text{opposite}}{\text{hypotenuse}}$

2. The Cosine Formula

The Cosine of an angle (written as $\cos x$) represents the ratio of the length of the adjacent side to the length of the hypotenuse.

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

3. The Tangent Formula

The Tangent of an angle (written as $\tan x$) represents the ratio of the opposite side to the adjacent side.

$\tan x = \frac{\text{opposite}}{\text{adjacent}}$

Remembering the Ratios: SOH CAH TOA

Because these three formulas are used so frequently, students often use a mnemonic device to memorize them. The most common one is SOH CAH TOA.

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

Study Tip: When you sit down for a test, the first thing you should do is scribble "SOH CAH TOA" in the margin of your paper. This prevents "brain freeze" from making you forget which side goes where during a difficult problem.

Worked Example

Let's put this into practice. Imagine you have a right-angled triangle where:

• The angle $x$ is $30^\circ$.
• The hypotenuse is $10\text{ cm}$ long.
• You want to find the length of the opposite side.

Step 1: Choose the correct ratio. Since we know the hypotenuse and we want to find the opposite, we look for the mnemonic that uses "O" and "H". That is SOH (Sine).

Step 2: Set up the equation. $\sin(30^\circ) = \frac{\text{opposite}}{10}$

Step 3: Solve for the unknown. We know that $\sin(30^\circ) = 0.5$. $0.5 = \frac{\text{opposite}}{10}$ Multiply both sides by $10$: $\text{opposite} = 0.5 \cdot 10$ $\text{opposite} = 5\text{ cm}$

By mastering these three ratios and the identification of the sides, you can solve a vast array of problems involving heights, distances, and angles in the real world.