Sine, Cosine and Tangents of angels

Introduction

In a right-angled triangle, the relationships between the acute angles and the lengths of the sides are called trigonometric ratios. The three primary ratios are Sine (sin), Cosine (cos), and Tangent (tan).

These ratios depend on the position of the reference angle ($\theta$):

• Hypotenuse (H): The longest side, directly opposite the $90^\circ$ angle.
• Opposite (O): The side facing the reference angle $\theta$.
• Adjacent (A): The side next to the reference angle $\theta$.

Key Points

We use the popular acronym SOH CAH TOA to remember the formulas:

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

Worked Example

If a right-angled triangle has an Opposite side of $3\text{ cm}$, an Adjacent side of $4\text{ cm}$, and a Hypotenuse of $5\text{ cm}$:

• $\sin(\theta) = \frac{3}{5} = 0.6$
• $\cos(\theta) = \frac{4}{5} = 0.8$
• $\tan(\theta) = \frac{3}{4} = 0.75$

Practical Activity

Real-life surveyors use these ratios to calculate the heights of tall mountains or buildings from a safe distance.

Home Practice: Measure your height and the length of your shadow outside, then use the tangent ratio ($\tan(\theta) = \frac{\text{Height}}{\text{Shadow}}$) to find the angle of elevation of the sun!