Chapter 22: Trigonometric Ratios

Lecture 1: Introduction to Trigonometric Ratios

Overview

• 6 trigonometric ratios:
  • sin(θ) (sinus)
  • cos(θ) (cosinus)
  • tan(θ) (tangent)
  • cot(θ) (cotangent)
  • sec(θ) (secant)
  • cosec(θ) (cosecant)
• Reciprocal relationships:
  • sin(θ) = 1 / cosec(θ)
  • cos(θ) = 1 / sec(θ)
  • tan(θ) = 1 / cot(θ)
  • cosec(θ) = 1 / sin(θ)
  • sec(θ) = 1 / cos(θ)
  • cot(θ) = 1 / tan(θ)

Right-Angle Triangle Components

• Hypotenuse: side opposite the right angle
• Perpendicular: side opposite the angle θ
• Base: the remaining side

Mnemonic for remembering ratios

• Pandit Badri Par Sona
  • P (sin θ) = Perpendicular / Hypotenuse
  • B (cos θ) = Base / Hypotenuse
  • T (tan θ) = Perpendicular / Base
  • Reciprocal Ratios:
    • cosec θ = Hypotenuse / Perpendicular
    • sec θ = Hypotenuse / Base
    • cot θ = Base / Perpendicular

Example Problems

Example 1

• Given sin(a) = 5/13, find tan(a)
  • Draw a right triangle
  • Let BC = 5k, AC = 13k
  • Use Pythagoras Theorem: AB² = AC² - BC²
    • AB = 12k
  • tan(a) = BC / AB = 5k / 12k = 5/12

Example 2

• Given tan(a) = 3/5, find sin(a)
  • Draw a right triangle
  • Let BC = 3k, AB = 5k
  • Use Pythagoras Theorem: AC² = BC² + AB²
    • AC = √34k
  • sin(a) = BC / AC = 3k / √34k = 3/√34

Example 3

• Find the value of cos²(a) + sin²(a)
  • cos²(a) + sin²(a) = 1 (by Pythagorean identity)

Example 4

• Given various triangles with sides, find the trigonometric ratios (sin, cos, tan) using the sides and Pythagorean theorem as needed.
  • Example: Given AB = 26, BD = 10, CD = 24, find tan(C)
    • Use Pythagorean theorem to find missing side.
    • tan(C) = Perpendicular / Base

Homework Problems

• Complete exercise 22A
  • Q1-Q15 dealing with various right triangles and trigonometric ratios. Use given sides and Pythagorean theorem to solve for unknowns.
  • Examples include solving for sin, cos, tan, and their reciprocals given sides of triangles.

Important Formulas

• sin(θ) = Perpendicular / Hypotenuse

• cos(θ) = Base / Hypotenuse

• tan(θ) = Perpendicular / Base

• cosec(θ) = Hypotenuse / Perpendicular

• sec(θ) = Hypotenuse / Base

• cot(θ) = Base / Perpendicular

• Pythagorean Theorem: In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

  • a² + b² = c²

Closing Remarks

• Practice drawing right triangles and labeling sides based on given information.
• Use mnemonic and formulas to solve problems quickly.
• Review homework problems and ensure understanding of using Pythagorean theorem in context of trigonometric ratios.

Next Steps

• Continue with exercise 22A to reinforce understanding.