Right Triangle Trigonometry Lecture Notes

Introduction

• Expression: SOH CAH TOA
  • Mnemonic for remembering trigonometric ratios in right triangles.

Components of a Right Triangle

• Sides of Triangle: Related to Angle Theta
  • Opposite: Side opposite to theta.
  • Adjacent: Side next to theta.
  • Hypotenuse: Longest side, opposite the right angle.

Pythagorean Theorem

• Equation: (a^2 + b^2 = c^2)
  • Application: Used for finding the length of a side in a right triangle.

Trigonometric Functions

• SOH CAH TOA

  • Sine (sin): Opposite over Hypotenuse ((\text{sin} \theta = \frac{opposite}{hypotenuse}))
  • Cosine (cos): Adjacent over Hypotenuse ((\text{cos} \theta = \frac{adjacent}{hypotenuse}))
  • Tangent (tan): Opposite over Adjacent ((\text{tan} \theta = \frac{opposite}{adjacent}))

• Reciprocal Functions

  • Cosecant (csc): (1/\text{sin}), i.e., (\frac{hypotenuse}{opposite})
  • Secant (sec): (1/\text{cos}), i.e., (\frac{hypotenuse}{adjacent})
  • Cotangent (cot): (1/\text{tan}), i.e., (\frac{adjacent}{opposite})

Examples and Special Right Triangles

• Example 1: Triangle with sides 3, 4, and hypotenuse 5

  • Finding Missing Side Using Pythagorean Theorem
    • (3^2 + 4^2 = 5^2 \rightarrow 9 + 16 = 25 \rightarrow c = 5)
  • Trigonometric Values:
    • (\sin \theta = \frac{4}{5})
    • (\cos \theta = \frac{3}{5})
    • (\tan \theta = \frac{4}{3})
    • (\csc \theta = \frac{5}{4})
    • (\sec \theta = \frac{5}{3})
    • (\cot \theta = \frac{3}{4})

• Special Triplets:

  • (3, 4, 5)
  • (5, 12, 13)
  • (8, 15, 17)
  • (7, 24, 25)
  • and multiples thereof.

Solving Right Triangles

• Example 2: Right triangle with sides 8, 17 and angle theta
  • Missing Side: 15 (using Pythagorean theorem)
  • Trigonometric Values:
    • (\sin \theta = \frac{15}{17})
    • (\cos \theta = \frac{8}{17})
    • (\tan \theta = \frac{15}{8})
    • (\csc \theta = \frac{17}{15})
    • (\sec \theta = \frac{17}{8})
    • (\cot \theta = \frac{8}{15})

Application of Trigonometric Functions

• Example 3: Given angle 38 degrees, side 42, find x

  • Use Tangent Function
    • (\tan 38 = \frac{x}{42})
    • (x = 42 \times \tan 38 \approx 32.8)

• Example 4: Angle 54 degrees, hypotenuse 26

  • Cosine Function
    • (\cos 54 = \frac{x}{26})
    • (x = 26 \times \cos 54 \approx 15.28)

Calculating Missing Angles

• Example 5: Find angle with sides 5 (opposite) and 4 (adjacent)
  • Use Tangent Function
    • (\tan \theta = \frac{5}{4})
    • (\theta = \tan^{-1} \left( \frac{5}{4} \right) \approx 51.34^\circ)

Course Information

• Trigonometry Course Available on Udemy
  • Topics Covered:
    • Angles, Unit Circle, Right Triangle Trigonometry, Graphing Functions, etc.
  • Features: Video quizzes, detailed breakdowns, practice problems.
    • Future Updates: Law of Sines, Law of Cosines, Polar Coordinates, etc.

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These notes summarize the key points about right triangle trigonometry discussed in the lecture, including practical examples, special triangles, trigonometric functions, and ways to solve related problems.