Lectures on Trigonometry

Introduction

• Subject: Trigonometry
• Focus: Right triangles and the relationships between their sides and angles.
• Objective: Using known sides/angles to find unknown sides/angles.

Basic Concepts

• Right Triangle: Composed of angles and sides with specific relationships.
• Angle Notation: The concerned angle is denoted by the symbol θ (theta).
• Side Ratios: The lengths of sides have specific ratios depending on the angle θ.

Sides of the Triangle

• Hypotenuse (h): The longest side in the triangle.
• Opposite (o): The side opposite to angle θ.
• Adjacent (a): The side next to angle θ.

Trigonometric Functions

• Sine (sin): Ratio of opposite to hypotenuse.
  • Formula: sin θ = opposite/hypotenuse
• Cosine (cos): Ratio of adjacent to hypotenuse.
  • Formula: cos θ = adjacent/hypotenuse
• Tangent (tan): Ratio of opposite to adjacent.
  • Formula: tan θ = opposite/adjacent

Mnemonic for Functions

• Phrase: "Old Sheikh Never Ever Breaks their Ancient Era"
  • sin = opposite/hypotenuse
  • cos = adjacent/hypotenuse
  • tan = opposite/adjacent

Solving Trigonometric Problems

1. Labeling Triangle Sides: Identify hypotenuse, opposite, and adjacent.
2. Determine the Function: Choose the proper trigonometric function based on known sides.

Sample Problems

Example 1

• Given Triangle: angle = 35°, hypotenuse = 12m, find opposite side.
• Function Used: Sine
  • sin 35° = x/12
  • Calculation: sin 35° = 0.57
  • Solve for x: x = 12 * 0.57
  • Result: Opposite side = 6.88m*

Example 2

• Given Triangle: angle = 48°, opposite = 15m, find adjacent side.
• Function Used: Tangent
  • tan 48° = 15/x
  • Calculation: tan 48° = 1.11
  • Solve for x: x = 15 / 1.11
  • Result: Adjacent side ≈ 13.51m

Finding Angles

• Given: Lengths of two sides.
• Process:
  1. Label the sides (hypotenuse, opposite).
  2. Use correct function (e.g., sin for opposite and hypotenuse).
  3. Calculate the ratio, and use the inverse function to find the angle.

Example 3

• Given: hypotenuse = 105m, opposite = 33m
• Function: Sine
  • sin θ = 33/105
  • Calculation: sin θ ≈ 0.314
  • Use inverse function: θ = sin⁻¹(0.314)
  • Result: θ ≈ 18.3°

Example 4

• Given: hypotenuse = 17, adjacent = 12
• Function: Cosine
  • cos θ = 12/17
  • Calculation: cos θ ≈ 0.71
  • Use inverse function: θ = cos⁻¹(0.71)
  • Result: θ ≈ 45.1°

Conclusion

• Trigonometry becomes simple with practice.
• Understanding calculator use is key to solving problems.
• Additional resources and help available (such as Patreon).
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