Right Triangle Trigonometry Lecture Notes

Introduction to SOHCAHTOA

• The expression SOHCAHTOA helps remember the definitions of sine, cosine, and tangent in right triangles.

Triangle Definitions

• Angle Theta (θ): Defined angle in the triangle.
• Sides of the Triangle:
  • Opposite Side: Opposite to angle θ.
  • Adjacent Side: Next to angle θ.
  • Hypotenuse: Longest side, opposite the right angle.

Pythagorean Theorem

• Basic equation: a² + b² = c² (where c is the hypotenuse).

Six Trigonometric Functions

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

Example Problem 1

• Given a right triangle with sides 3 and 4:
  • Use Pythagorean theorem to find hypotenuse (c):
    • 3² + 4² = 9 + 16 = 25 → c = √25 = 5.
  • Trigonometric Values:
    • sin(θ) = 4/5
    • cos(θ) = 3/5
    • tan(θ) = 4/3
    • csc(θ) = 5/4
    • sec(θ) = 5/3
    • cot(θ) = 3/4

Special Right Triangles

• Common Ratios:
  • 3-4-5 triangle
  • 5-12-13 triangle
  • 8-15-17 triangle
  • 7-24-25 triangle
• Whole number multiples also apply (e.g., 6-8-10, 9-12-15).
• Less common triangles include 9-40-41 and 1160-61.

Example Problem 2

• Given sides 8 and 17:
  • Use Pythagorean theorem:
    • 8² + b² = 17² → 64 + b² = 289 → b = 15.
  • Trigonometric Values:
    • sin(θ) = 15/17
    • cos(θ) = 8/17
    • tan(θ) = 15/8
    • csc(θ) = 17/15
    • sec(θ) = 17/8
    • cot(θ) = 8/15

Example Problem 3

• Given hypotenuse 25 and side 15:
  • The missing side is 20 (3-4-5 triangle ratios apply).
  • Trigonometric Values:
    • sin(θ) = 20/25 = 4/5
    • cos(θ) = 15/25 = 3/5
    • tan(θ) = 20/15 = 4/3
    • csc(θ) = 5/4
    • sec(θ) = 5/3
    • cot(θ) = 3/4

Finding Missing Sides/Angles

• Use appropriate trig functions based on given sides relative to the angle.
• Example: For θ = 38° and adjacent = 42, use tangent:
  • x = 42 tan(38°) → x ≈ 32.8.

Finding Angles

• Given two sides, use inverse functions:
  • For opposite = 5 and adjacent = 4:
    • θ = arctan(5/4) → θ ≈ 51.34°.
• Example: Opposite = 5, Hypotenuse = 6:
  • θ = arcsin(5/6) → θ ≈ 56.44°.

Course Information

• The trigonometry course covers:
  • Angles and radians
  • Unit circle and trig functions
  • Right triangle trigonometry
  • Angle elevation/depression problems
  • Inverse trig functions and their applications
  • Graphing trig functions
  • Trig identities and equations
  • Special topics like Law of sines and cosines

Conclusion

• Understanding right triangle trigonometry is essential for solving geometric problems and applying trigonometric functions effectively.