Trigonometric Ratios and SOHCAHTOA

Introduction

• The acronym SOHCAHTOA helps remember trigonometric ratios in right triangles:
  • S: Sine = Opposite / Hypotenuse
  • C: Cosine = Adjacent / Hypotenuse
  • T: Tangent = Opposite / Adjacent
• Useful for solving problems involving right triangles.

Understanding Trigonometric Ratios

• Sine of an angle is the ratio of the opposite side to the hypotenuse.
• Cosine of an angle is the ratio of the adjacent side to the hypotenuse.
• Tangent of an angle is the ratio of the opposite side to the adjacent side.
• Applicable only to right triangles.
• Only acute angles (non-right angles) are used to find sine, cosine, and tangent.

Example Problems

Example 1: Triangle ABC

• Given: Right triangle with angles A, B, and C.
• Objective: Find sine, cosine, and tangent for angles A and B.
• Angle A:
  • Sine(A) = Opposite / Hypotenuse = 5/13
  • Cosine(A) = Adjacent / Hypotenuse = 12/13
  • Tangent(A) = Opposite / Adjacent = 5/12
• Angle B:
  • Sine(B) = Opposite / Hypotenuse = 12/13
  • Cosine(B) = Adjacent / Hypotenuse = 5/13
  • Tangent(B) = Opposite / Adjacent = 12/5

Example 2: Solving for a Side

• Given: Right triangle with a 40-degree angle, opposite side = 5, adjacent side = X.
• Objective: Find the length of the missing side X using tangent.
• Solution:
  • Use Tangent(40°) = Opposite / Adjacent
  • Tangent(40°) = 5/X
  • Cross multiply to find X.
  • Calculate using a calculator.

Example 3: Cosine Ratio

• Given: Right triangle with 35-degree angle, adjacent side = X, hypotenuse = 8.
• Objective: Find the length of the missing side X using cosine.
• Solution:
  • Cosine(35°) = Adjacent / Hypotenuse = X/8
  • Cross multiply to solve for X.
  • Calculate using a calculator.

Example 4: Sine Ratio

• Given: Right triangle with 50-degree angle, opposite side = 4, hypotenuse = Y.
• Objective: Find the length of the missing side Y using sine.
• Solution:
  • Sine(50°) = Opposite / Hypotenuse = 4/Y
  • Use technique of switching diagonal terms to simplify calculation.
  • Calculate using a calculator to find Y.

Important Tips

• Use SOHCAHTOA for remembering which sides to use for sine, cosine, and tangent.
• Always use acute angles for trigonometric ratio calculations, not the right angle.
• Techniques include cross-multiplying and switching diagonals in proportions for solving equations.

Conclusion

• These trigonometric ratios are essential for solving problems involving right triangles.
• Helps determine missing sides or angles when certain values are known.