Right Triangle Trigonometry Review

Basic Trigonometric Ratios

• Sine (sin): Opposite side over hypotenuse
• Cosine (cos): Adjacent side over hypotenuse
• Tangent (tan): Opposite side over adjacent

Mnemonic

• SOH-CAH-TOA:
  • Sine (sin) = Opposite / Hypotenuse
  • Cosine (cos) = Adjacent / Hypotenuse
  • Tangent (tan) = Opposite / Adjacent

Pythagorean Theorem

• Formula: (a^2 + b^2 = c^2)
  • (a) and (b) are the legs of the triangle
  • (c) is the hypotenuse

Reciprocal Trigonometric Functions

• Cosecant (csc): Reciprocal of sine, Hypotenuse over opposite
• Secant (sec): Reciprocal of cosine, Hypotenuse over adjacent
• Cotangent (cot): Reciprocal of tangent, Adjacent over opposite

Right Triangle Components

• Adjacent Side: Next to the angle
• Opposite Side: Across from the angle
• Hypotenuse: Longest side, across from the 90° angle

Evaluating Trig Functions

1. Given sides: Identify opposite, adjacent, hypotenuse
2. Find the functions:
   • (\text{sin}\theta = \frac{\text{opposite}}{\text{hypotenuse}})
   • (\text{cos}\theta = \frac{\text{adjacent}}{\text{hypotenuse}})
   • (\text{tan}\theta = \frac{\text{opposite}}{\text{adjacent}})
3. Reciprocal functions:
   • (\text{csc}\theta = \frac{\text{hypotenuse}}{\text{opposite}})
   • (\text{sec}\theta = \frac{\text{hypotenuse}}{\text{adjacent}})
   • (\text{cot}\theta = \frac{\text{adjacent}}{\text{opposite}})
4. Example:
   • Given opposite = 12, hypotenuse = 13:
     • (\text{sin} = \frac{12}{13})
     • (\text{csc} = \frac{13}{12})

Solving for Hypotenuse

• Using Pythagorean Theorem: Solve for (c)
  • (a^2 + b^2 = c^2)
  • Example: (10^2 + 5^2 = c^2)
  • Solve to find (c)

Solving Right Triangles

• Finding Missing Angles:

  • Sum of angles = 180°
  • Right angle = 90°
  • Remaining angles sum to 90°

• Using Trigonometric Functions:

  • Use known side lengths and one angle
  • Examples:
    • (\text{tan}62° = \frac{b}{6})
    • Solve for missing sides

Application Problems

• Example: Flagpole height
  • Given distance and angle of elevation
  • Use tangent: (\text{tan}65° = \frac{\text{height}}{20})
  • Solve for height

Tips

• No decimals or mixed numbers for trigonometric ratios
• Practice: Aligning trigonometric ratios correctly
• Ensure calculator is in degree mode for calculations

Note: Ensure understanding of reciprocal functions and practice with various triangle problems for mastery.