Sine, Cosine and Tangent

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Introduction

The three fundamental trigonometric ratios - sine, cosine, and tangent - are the building blocks of trigonometry. They relate the angles of a right-angled triangle to the ratios of its sides.

Definitions

In a right-angled triangle with angle θ:

  • sin θ = opposite/hypotenuse
  • cos θ = adjacent/hypotenuse
  • tan θ = opposite/adjacent

Note: tan θ = sin θ/cos θ

Example 1: Finding Trigonometric Ratios

In a right-angled triangle, if the opposite side is 3, adjacent side is 4, and hypotenuse is 5, find sin θ, cos θ, and tan θ

Solution:

sin θ = opposite/hypotenuse = 3/5 = 0.6

cos θ = adjacent/hypotenuse = 4/5 = 0.8

tan θ = opposite/adjacent = 3/4 = 0.75

Answer: sin θ = 0.6, cos θ = 0.8, tan θ = 0.75

Example 2: Using Trigonometric Ratios to Find Sides

In a right-angled triangle, if θ = 30° and the hypotenuse is 10, find the opposite and adjacent sides

Solution:

sin 30° = opposite/hypotenuse

0.5 = opposite/10

opposite = 10 × 0.5 = 5

cos 30° = adjacent/hypotenuse

0.866 = adjacent/10

adjacent = 10 × 0.866 = 8.66

Answer: opposite = 5, adjacent = 8.66

Example 3: Finding Angles

In a right-angled triangle, if the opposite side is 6 and adjacent side is 8, find angle θ

Solution:

tan θ = opposite/adjacent = 6/8 = 0.75

θ = tan⁻¹(0.75) = 36.87°

Answer: θ = 36.87°

Key Properties

  • Range: sin θ and cos θ are between -1 and 1
  • tan θ: can be any real number
  • Pythagorean Identity: sin²θ + cos²θ = 1
  • Quotient Identity: tan θ = sin θ/cos θ
  • Reciprocal Identities: csc θ = 1/sin θ, sec θ = 1/cos θ, cot θ = 1/tan θ

Example 4: Using the Pythagorean Identity

If sin θ = 0.6, find cos θ

Solution:

Using sin²θ + cos²θ = 1:

(0.6)² + cos²θ = 1

0.36 + cos²θ = 1

cos²θ = 1 - 0.36 = 0.64

cos θ = ±√0.64 = ±0.8

Answer: cos θ = ±0.8

Key Points to Remember

  • Always identify the opposite, adjacent, and hypotenuse sides
  • Use SOHCAHTOA to remember the definitions
  • Check that your answers are reasonable
  • Use the Pythagorean identity to verify results
  • Remember the ranges of the trigonometric functions

Problem-Solving Strategy

  1. Draw a diagram and label the sides
  2. Identify which trigonometric ratio to use
  3. Set up the equation
  4. Solve for the unknown
  5. Check your answer is reasonable