Right Triangle Trigonometry

SOH CAH TOA

• Definition: A mnemonic to remember the definitions of sine, cosine, and tangent.
• Sine (SOH): ( \sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}} )
• Cosine (CAH): ( \cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}} )
• Tangent (TOA): ( \tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} )

Triangle Sides

• Opposite Side: side opposite to angle ( \theta )
• Adjacent Side: side next to angle ( \theta )
• Hypotenuse: longest side, opposite the right angle

Pythagorean Theorem

• Formula: ( a^2 + b^2 = c^2 )
• A reminder for right triangles; not the main focus of this lesson.

Six Trigonometric Functions

1. Sine: ( \sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}} )
2. Cosine: ( \cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}} )
3. Tangent: ( \tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} )
4. Cosecant: ( \csc(\theta) = \frac{1}{\sin(\theta)} = \frac{\text{Hypotenuse}}{\text{Opposite}} )
5. Secant: ( \sec(\theta) = \frac{1}{\cos(\theta)} = \frac{\text{Hypotenuse}}{\text{Adjacent}} )
6. Cotangent: ( \cot(\theta) = \frac{1}{\tan(\theta)} = \frac{\text{Adjacent}}{\text{Opposite}} )

Example Problem 1

Given:

• One side = 3, other side = 4
• Find hypotenuse using Pythagorean theorem:
  • ( 3^2 + 4^2 = c^2 )
  • ( 9 + 16 = 25 )
  • ( c = 5 )

Finding Trigonometric Functions:

• Sine: ( \sin(\theta) = \frac{4}{5} )
• Cosine: ( \cos(\theta) = \frac{3}{5} )
• Tangent: ( \tan(\theta) = \frac{4}{3} )
• Cosecant: ( \csc(\theta) = \frac{5}{4} )
• Secant: ( \sec(\theta) = \frac{5}{3} )
• Cotangent: ( \cot(\theta) = \frac{3}{4} )

Special Right Triangles

• Common Triples:
  • 3, 4, 5
  • 5, 12, 13
  • 8, 15, 17
  • 7, 24, 25

Multiples of Special Triples

• Multiply by integers (e.g., 2, 3) to create new triangles.

Example Problem 2

Given:

• Sides = 8, 17 (Find missing side)
• Recognize as 8-15-17 triangle.
• Missing side = 15 using:
  • ( 8^2 + 15^2 = 17^2 )

Finding Trigonometric Functions:

• Sine: ( \sin(\theta) = \frac{15}{17} )
• Cosine: ( \cos(\theta) = \frac{8}{17} )
• Tangent: ( \tan(\theta) = \frac{15}{8} )
• Cosecant: ( \csc(\theta) = \frac{17}{15} )
• Secant: ( \sec(\theta) = \frac{17}{8} )
• Cotangent: ( \cot(\theta) = \frac{8}{15} )

Example Problem 3

• Given hypotenuse = 25, side = 15.
• Recognize as 15-20-25 triangle.
• Find missing side (20) and functions:
  • Sine: ( \sin(\theta) = \frac{20}{25} = \frac{4}{5} )
  • Cosine: ( \cos(\theta) = \frac{15}{25} = \frac{3}{5} )
  • Tangent: ( \tan(\theta) = \frac{20}{15} = \frac{4}{3} )
  • Cosecant: ( \csc(\theta) = \frac{5}{4} )
  • Secant: ( \sec(\theta) = \frac{5}{3} )
  • Cotangent: ( \cot(\theta) = \frac{3}{4} )

Finding Missing Sides and Angles

General Steps:

• Identify known sides relative to angle.
• Use appropriate trigonometric function (sine, cosine, tangent).
• Rearrange to isolate the unknown variable.

Example: Given Opposite Side x and Adjacent Side 42

• Use tangent: ( \tan(38^{\circ}) = \frac{x}{42} )
• Solve for x: ( x = 42 \tan(38^{\circ}) )
• Get x using calculator (ensure in degree mode).

Example: Given Hypotenuse = 26, Adjacent Side X

• Use cosine: ( \cos(54^{\circ}) = \frac{x}{26} )
• Solve for x: ( x = 26 \cos(54^{\circ}) )

Finding Angles

• Use inverse trig functions (e.g., ( \theta = \tan^{-1} \left( \frac{5}{4} \right) ))

Conclusion

• Trigonometric functions help solve right triangle problems efficiently.
• Practice using different examples to gain proficiency.

Course Access

• Trigonometry Course: Available on Udemy
• Topics: Angles, Unit Circle, Right Triangle Trigonometry, Applications, Graphing, Inverse Functions, Trig Identities, etc.