Solving Right Triangle Problems Using SOH-CAH-TOA

Introduction

• Today's topic is solving for the missing side of a right triangle using trigonometric ratios.
• The focus is on SOH-CAH-TOA.

Trigonometric Ratios

1. SOH (Sine)
   • Sine = Opposite / Hypotenuse
2. CAH (Cosine)
   • Cosine = Adjacent / Hypotenuse
3. TOA (Tangent)
   • Tangent = Opposite / Adjacent

Example 1: Solving with Sine (SOH)

• Given: Right triangle with an acute angle of 36°.
  • Hypotenuse = 10 cm
  • Missing side (opposite) = X
• Approach:
  • Identify that the side opposite the 36° angle is the side X.
  • Recognize the hypotenuse as 10 cm.
  • Use the sine ratio:
    • ( \sin(36°) = \frac{X}{10} )
  • Cross-multiply:
    • ( 1 \times X = 10 \times \sin(36°) )
  • Calculate using a calculator:
    • ( \sin(36°) \approx 0.5878 )
    • ( X \approx 10 \times 0.5878 = 5.878 )
  • Round to one decimal place:
    • ( X \approx 5.9 \text{ cm} )

Example 2: Solving with Tangent (TOA)

• Given: Right triangle with an acute angle of 40°.
  • Adjacent side = 6 cm
  • Missing side (opposite) = X
• Approach:
  • Identify the side opposite the 40° angle as X.
  • Recognize the adjacent side as 6 cm.
  • Use the tangent ratio:
    • ( \tan(40°) = \frac{X}{6} )
  • Cross-multiply:
    • ( 1 \times X = 6 \times \tan(40°) )
  • Calculate using a calculator:
    • ( \tan(40°) \approx 0.8391 )
    • ( X \approx 6 \times 0.8391 = 5.0346 )
  • Round to one decimal place:
    • ( X \approx 5 \text{ cm} ) (when rounded to the nearest whole number)

Conclusion

• Reviewed how to solve for missing sides in right triangles using sine and tangent ratios.
• Encouraged practice by solving a new problem:
  • Given: Right triangle with a 34° angle and one side = 12 cm.
  • Find the missing side.

Additional Notes

• Importance of using a scientific calculator for trigonometric calculations.
• Emphasized the rounding of answers to maintain consistency in decimal places.

Assignment: Solve the new problem and post answers in the comments.