TechMath Lecture: Introduction to Trigonometry

Overview

• Trigonometry is a branch of mathematics dealing with the relationships between the sides and angles in right-angled triangles.
• Allows calculation of unknown side lengths and angles using known values.

Key Concepts

Triangle and Angle Notation

• Right angle triangle: contains a 90-degree angle.
• Important parts:
  • Theta (θ): A common notation for an angle in a triangle.
  • Hypotenuse (H): The longest side opposite the right angle.
  • Opposite (O): The side opposite to the angle theta.
  • Adjacent (A): The side next to the angle theta.

Trigonometric Functions

• Sine (sin): Ratio of the opposite side to the hypotenuse.
• Cosine (cos): Ratio of the adjacent side to the hypotenuse.
• Tangent (tan): Ratio of the opposite side to the adjacent side.

Mnemonic for Remembering Functions

• "Some Old Hags Can’t Always Hack Their Old Age":
  • SOH: Sine = Opposite / Hypotenuse
  • CAH: Cosine = Adjacent / Hypotenuse
  • TOA: Tangent = Opposite / Adjacent

Example Problems

Example 1: Finding Unknown Side Length

• Given a triangle with:
  • Angle of 35 degrees
  • One side (hypotenuse) of 12 meters
• Use sine function: sin(theta) = Opposite / Hypotenuse
  • Setup equation: sin(35) = x / 12
  • Solve using calculator: sin(35) ≈ 0.57
  • Solve for x: x = 12 * 0.57 = 6.88 meters

Example 2: Another Side Length Problem

• Given:
  • Angle of 48 degrees
  • Opposite side length of 15 meters
• Use tangent function: tan(theta) = Opposite / Adjacent
  • Setup: tan(48) = 15 / x
  • Solve for x: x = 15 / tan(48) ≈ 13.51 meters

Example 3: Finding the Angle

• Known side lengths:
  • Opposite: 33 meters
  • Hypotenuse: 105 meters
• Use sine function: sin(theta) = Opposite / Hypotenuse
  • Setup: sin(theta) = 33 / 105 ≈ 0.314
  • Find angle: theta = sin^(-1)(0.314) ≈ 18.3 degrees

Example 4: Another Angle Calculation

• Known side lengths:
  • Adjacent: 12
  • Hypotenuse: 17
• Use cosine function: cos(theta) = Adjacent / Hypotenuse
  • Setup: cos(theta) = 12 / 17 ≈ 0.71
  • Find angle: theta = cos^(-1)(0.71) ≈ 45.1 degrees

Conclusion

• Trigonometry can be used for solving problems involving right-angled triangles by understanding and applying the trigonometric functions.
• Familiarity with using a calculator for trigonometric functions and their inverses is essential.

Additional Resources

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