Trigonometry Basics

Overview

This lecture introduces the basics of trigonometry, focusing on right-angled triangles and the relationships between their sides and angles.

Introduction to Trigonometry

• Trigonometry is the study of relationships between side lengths and angles in right-angled triangles.
• Knowing one angle and one side allows you to calculate other sides; knowing two sides allows calculation of unknown angles.

Triangle Side Labeling

• The longest side, opposite the right angle, is called the hypotenuse (H).
• The side opposite the angle of interest (Theta, θ) is called the opposite (O).
• The side next to the angle of interest, not the hypotenuse, is called the adjacent (A).

Main Trigonometric Functions

• Sine (sin θ) = Opposite / Hypotenuse.
• Cosine (cos θ) = Adjacent / Hypotenuse.
• Tangent (tan θ) = Opposite / Adjacent.
• Mnemonic: “Some Old Hags Can’t Always Hack Their Old Age” helps remember the ratios.

Solving for Unknown Side (Example 1)

• Label triangle sides according to angle θ.
• If angle is 35°, hypotenuse is 12 m, and opposite is unknown, use sin 35° = x/12.
• Calculate sin 35° ≈ 0.57, so x = 12 × 0.57 = 6.88 m.

Solving for Unknown Side (Example 2)

• Given angle 48°, opposite side 15 m, adjacent side unknown, use tan 48° = 15/x.
• Calculate tan 48° ≈ 1.11, so x = 15 / 1.11 = 13.51 m.

Solving for Unknown Angle (Example 1)

• Given hypotenuse 105 m, opposite 33 m, find angle θ with sin θ = 33/105 ≈ 0.314.
• Use inverse sine (sin⁻¹) on calculator: θ = sin⁻¹(0.314) ≈ 18.3°.

Solving for Unknown Angle (Example 2)

• Given hypotenuse 17, adjacent 12, find angle θ with cos θ = 12/17 ≈ 0.71.
• Use inverse cosine (cos⁻¹) on calculator: θ = cos⁻¹(0.71) ≈ 45.1°.

Key Terms & Definitions

• Hypotenuse — the longest side opposite the right angle in a right triangle.
• Opposite — the side directly across from the angle of interest.
• Adjacent — the side next to the angle of interest (but not the hypotenuse).
• Sine (sin θ) — ratio of opposite side to hypotenuse.
• Cosine (cos θ) — ratio of adjacent side to hypotenuse.
• Tangent (tan θ) — ratio of opposite side to adjacent side.
• Inverse function (sin⁻¹, cos⁻¹) — calculates the angle from a given trigonometric ratio.

Action Items / Next Steps

• Practice labeling triangle sides and setting up equations using sine, cosine, and tangent.
• Familiarize yourself with using inverse trigonometric functions on your calculator.
• Try solving example problems by calculating unknown sides and angles in right-angled triangles.