Sine, Cosine, and Tangent

Introduction to Trigonometric Functions

• Sine (sin), Cosine (cos), and Tangent (tan) are key functions in trigonometry.
• Based on a right-angled triangle.
• Sides of a triangle:
  • Opposite: Opposite to the angle.
  • Adjacent: Next to the angle.
  • Hypotenuse: Longest side.

Calculating Trigonometric Ratios

• Ratios are constant for a given angle, regardless of triangle size.
• Formulae:
  • Sine (sin): Opposite/Hypotenuse
  • Cosine (cos): Adjacent/Hypotenuse
  • Tangent (tan): Opposite/Adjacent

Example Calculations

• Sine of 35°:
  • sin(35°) = Opposite/Hypotenuse = 2.8/4.9 = 0.57
• Cosine of 35°:
  • cos(35°) = Adjacent/Hypotenuse = 4.0/4.9 = 0.82
• Tangent of 35°:
  • tan(35°) = Opposite/Adjacent = 2.8/4.0 = 0.70

Importance of Triangle Size

• Size of triangle doesn’t affect ratios, only the angle does.
• Visual aids: Mouse over to see effects of angle changes.

Memory Aid: SOHCAHTOA

• Soh: Sine = Opposite/Hypotenuse
• Cah: Cosine = Adjacent/Hypotenuse
• Toa: Tangent = Opposite/Adjacent

Angles and the Unit Circle

• Understanding angles from 0 to 360 degrees.
• Effects of angle on sine, cosine, and tangent values.
• Unit Circle: Hypotenuse is 1.

Examples with Common Angles

• 30° Triangle:
  • sin(30°) = 1/2 = 0.5
  • cos(30°) = √3/2 = 0.866
  • tan(30°) = 1/√3 = 0.577
• 45° Triangle:
  • sin(45°) = 1/√2 = 0.707
  • cos(45°) = 1/√2 = 0.707
  • tan(45°) = 1

Real-World Application

• Solving problems by finding unknown sides or angles.
• Example:
  • Problem: Find "d" in a triangle with a given angle and hypotenuse.
  • Steps: Use sin function and algebra to solve for "d."

Exercises

• Paper-based exercises with angles from 0 to 360 degrees.

Less Common Functions

• Secant (sec): 1/cos = Hypotenuse/Adjacent
• Cosecant (csc): 1/sin = Hypotenuse/Opposite
• Cotangent (cot): 1/tan = Adjacent/Opposite

Note: Practice regularly and use a calculator to ensure accuracy in calculations.