Introduction to Trigonometry (Grade 10)

Understanding the Right-Angled Triangle

• Right-Angle Triangle: The focus of trigonometry at this level.
• Greek Letters: Commonly used to denote angles, such as ( \theta ).
• Sides of a Triangle:
  • Hypotenuse: Opposite the right angle, next to angle ( \theta ).
  • Adjacent Side: Lies next to angle ( \theta ).
  • Opposite Side: Lies opposite angle ( \theta ).

Trigonometric Ratios

• Three Key Ratios:
  • Sine (sin ( \theta )) = Opposite / Hypotenuse
  • Cosine (cos ( \theta )) = Adjacent / Hypotenuse
  • Tangent (tan ( \theta )) = Opposite / Adjacent
• Mnemonic to Remember: "Oh Heck Another Hour of Algebra" (OHA – Opposite, Hypotenuse, Adjacent).
• Consistency Across Triangles: Ratios remain the same for identical angles regardless of triangle size.

Example Problems

Example 1

• Given: A triangle with sides identified.
• Steps to Solve:
  1. Identify sides: Hypotenuse, Adjacent, Opposite.
  2. Calculate ratios:
     • Sine ( \theta ): 6/10 = 0.6
     • Cosine ( \theta ): 8/10 = 0.8
     • Tangent ( \theta ): 6/8 = 0.75

Example 2

• Given: A triangle with missing side length.
• Steps:
  1. Use the Pythagorean theorem to find the missing side length.
  2. Calculate ratios:
     • Sine ( \theta ): 8/17 = 0.5
     • Cosine ( \theta ): 15/17 = 0.9
     • Tangent ( \theta ): 8/15 = 0.5

Trigonometry on the Cartesian Plane

Using a Point P

• Triangle Construction:
  • Point P: In the first quadrant.
  • Drawing the Triangle: From P, draw a line perpendicular to the x-axis.
  • Side Lengths:
    • X-axis length = x
    • Y-axis length = y
    • Hypotenuse (R) = Connects origin to P.
• Trig Ratios:
  • Sine ( \theta ): ( y/R )
  • Cosine ( \theta ): ( x/R )
  • Tangent ( \theta ): ( y/x )

Example with Point (3, 4)

• Construct Triangle:
  • Use Pythagorean theorem to find R = 5.
• Calculate Ratios:
  • Sine ( \theta ): 4/5 = 0.8
  • Cosine ( \theta ): 3/5 = 0.6
  • Tangent ( \theta ): 4/3 = 1.3

Conclusion

• Self-Test: Practice identifying and calculating trigonometric ratios using given angles and side lengths.