Lecture Notes: Right Triangle Trigonometry and Trigonometric Functions

Introduction to SOHCAHTOA and Right Triangle Trigonometry

• SOHCAHTOA: Mnemonic for remembering trigonometric ratios.
  • Sine: Opposite over Hypotenuse
  • Cosine: Adjacent over Hypotenuse
  • Tangent: Opposite over Adjacent
• Right Triangle Components:
  • Opposite side: Side opposite to the angle theta.
  • Adjacent side: Side next to the angle theta.
  • Hypotenuse: Longest side across the right angle.

Pythagorean Theorem

• Equation: (a^2 + b^2 = c^2) applies to right triangles.

Six Trigonometric Functions

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

Special Right Triangles and Pythagorean Triples

• Common Pythagorean triples:
  • 3-4-5, 5-12-13, 8-15-17, 7-24-25, 9-40-41, 11-60-61
• Multiples of these triples also work (e.g., 6-8-10, 9-12-15).

Solving Right Triangle Problems

Example Problem 1

• Given sides: 3 and 4, find hypotenuse and trigonometric functions.
• Calculate hypotenuse using Pythagorean theorem: (5).
• Calculate:
  • Sine: 4/5
  • Cosine: 3/5
  • Tangent: 4/3
  • Cosecant: 5/4
  • Secant: 5/3
  • Cotangent: 3/4

Example Problem 2

• Given sides: 8 and 17, find the missing side and trigonometric functions.
• Identify as part of 8-15-17 right triangle.
• Calculate:
  • Sine: 15/17
  • Cosine: 8/17
  • Tangent: 15/8
  • Cosecant: 17/15
  • Secant: 17/8
  • Cotangent: 8/15

Example Problem 3

• Using angle (theta) 38 degrees and side 42 to find x using tangent.
• Calculation: (x = 42 * \tan(38) = 32.8)

Example Problem 4

• Using angle 54 degrees and hypotenuse 26 to find adjacent side using cosine.
• Calculation: (x = 26 * \cos(54) = 15.28)

Example Problem 5

• Using angle 32 degrees, opposite side 12, and hypotenuse x using sine.
• Calculation: (x = 12 / \sin(32) = 22.64)

Finding Angles Using Inverse Trigonometric Functions

Example Problem 6

• Using sides 5 (opposite) and 4 (adjacent) to find angle using tangent.
• Calculation: (\theta = \arctan(5/4) = 51.34)

Example Problem 7

• Using sides 3 (adjacent) and 7 (hypotenuse) to find angle using cosine.
• Calculation: (\theta = \arccos(3/7) = 64.62)

Example Problem 8

• Using sides 5 (opposite) and 6 (hypotenuse) to find angle using sine.
• Calculation: (\theta = \arcsin(5/6) = 56.44)

Trigonometry Course Information

• Course on Udemy covering various trigonometry topics, including angles, unit circle, special triangles, and graphing functions.
• Emphasis on verifying trig identities, solving equations, and using formulas like law of sines and cosines.
• Course development ongoing with additional sections to be added.