Introduction to Trigonometric Ratios

What is Trigonometry?

Trigonometry is the branch of mathematics that studies the relationship between the angles and sides of triangles, particularly right-angled triangles. The word comes from Greek words meaning "triangle measurement".

The Right-Angled Triangle

In a right-angled triangle, relative to a chosen angle θ (theta, not the right angle), the three sides are named:

• Hypotenuse: the longest side, opposite the right angle.
• Opposite: the side directly across from angle θ.
• Adjacent: the side next to angle θ (that is not the hypotenuse).

The Three Basic Trigonometric Ratios

These ratios relate the angle θ to the ratio of two sides. They are remembered using the mnemonic SOH-CAH-TOA:

• Sine (sin θ) = Opposite / Hypotenuse (SOH)
• Cosine (cos θ) = Adjacent / Hypotenuse (CAH)
• Tangent (tan θ) = Opposite / Adjacent (TOA)

Common Trigonometric Values

Certain angles have well-known ratio values used frequently in calculations: sin 30° = 0.5, sin 45° ≈ 0.707, sin 60° ≈ 0.866, sin 90° = 1. cos 0° = 1, cos 30° ≈ 0.866, cos 60° = 0.5, cos 90° = 0. tan 45° = 1.

Using Trigonometry to Find Sides and Angles

If you know one angle (other than the right angle) and one side of a right-angled triangle, you can find the other sides using the appropriate ratio. For example, if the hypotenuse is 10 cm and angle θ = 30°, then the opposite side = 10 × sin 30° = 10 × 0.5 = 5 cm.

Real-Life Applications

Trigonometry is used in construction (calculating heights and slopes), navigation (determining direction and distance), engineering, physics, and even in designing video game graphics. Surveyors use it to measure land, and architects use it to design safe, stable structures.