Right Triangle Trigonometry

Introduction to SOH CAH TOA

• SOH CAH TOA refers to the relationships between the angles and sides of right triangles.
  • Sine (SOH): Sine θ = Opposite/Hypotenuse
  • Cosine (CAH): Cosine θ = Adjacent/Hypotenuse
  • Tangent (TOA): Tangent θ = Opposite/Adjacent

Triangle Sides

• Opposite side: The side opposite angle θ.
• Adjacent side: The side next to angle θ.
• Hypotenuse: The longest side, opposite the right angle.

Pythagorean Theorem

• The relationship of the sides of a right triangle:
  • A² + B² = C²

Six Trigonometric Functions

• Sine: ( ext{sine } θ = rac{ ext{Opposite}}{ ext{Hypotenuse}} )
• Cosine: ( ext{cosine } θ = rac{ ext{Adjacent}}{ ext{Hypotenuse}} )
• Tangent: ( ext{tangent } θ = rac{ ext{Opposite}}{ ext{Adjacent}} )
• Cosecant: ( ext{cosecant } θ = rac{ ext{Hypotenuse}}{ ext{Opposite}} )
• Secant: ( ext{secant } θ = rac{ ext{Hypotenuse}}{ ext{Adjacent}} )
• Cotangent: ( ext{cotangent } θ = rac{ ext{Adjacent}}{ ext{Opposite}} )

Special Right Triangles

• Recognized triangles and their side ratios:
  • 3, 4, 5 triangle
  • 5, 12, 13 triangle
  • 8, 15, 17 triangle
  • 7, 24, 25 triangle
• Multiples of these ratios also apply (e.g., 6, 8, 10).

Example Problem

• Given sides 3 and 4, find the hypotenuse:
  • Using Pythagorean Theorem: ( 3² + 4² = 5² )
• Calculate trigonometric functions:
  • Sine θ: ( rac{4}{5} )
  • Cosine θ: ( rac{3}{5} )
  • Tangent θ: ( rac{4}{3} )
  • Cosecant: ( rac{5}{4} )
  • Secant: ( rac{5}{3} )
  • Cotangent: ( rac{3}{4} )

Additional Problems

• Second triangle with sides 8 and 17:
  • Use Pythagorean Theorem: 8² + B² = 17²
  • Find missing side and calculate all six trig functions.
• Another example with hypotenuse 25 and side 15:
  • Use known ratios to find the missing side (20).

Finding Unknown Sides and Angles

• When given one angle and one side, identify which trig function to use (sine, cosine, or tangent).
• To isolate x or θ in equations, use algebraic manipulation and calculators for angles.

Calculator Use

• Ensure calculator is in degree mode for angle calculations.
• Use inverse functions (arcsin, arccos, arctan) to find angles based on sides.

Accessing Trigonometry Course

• Course available on Udemy covering:
  • Angles, radians, unit circle, right triangle trig, graphing functions, inverse trig functions, and more.
• Future updates to include other topics like the law of sines, law of cosines, and polar coordinates.