Right Triangle Trigonometry

SOH CAH TOA

• SOH: Sine = Opposite / Hypotenuse
• CAH: Cosine = Adjacent / Hypotenuse
• TOA: Tangent = Opposite / Adjacent

Triangle Components

• Theta (θ): The angle of reference.
• Opposite Side: Side opposite to θ.
• Adjacent Side: Side next to θ.
• Hypotenuse: The longest side of the right triangle, across from the right angle.

Pythagorean Theorem

• Formula: A² + B² = C²
• Used to find missing side lengths in right triangles.

Trigonometric Functions

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

Special Right Triangles

• 3-4-5 Triangle: Ratio of sides is 3:4:5.
• 5-12-13 Triangle: Ratio of sides is 5:12:13.
• 8-15-17 Triangle: Ratio of sides is 8:15:17.
• 7-24-25 Triangle: Ratio of sides is 7:24:25.
• Multiples of these ratios also yield valid triangles.

Example Problem 1

Given:

• Opposite = 4
• Adjacent = 3
• Hypotenuse (found using Pythagorean theorem): 5

Calculate Trigonometric Functions:

• sin θ = 4/5
• cos θ = 3/5
• tan θ = 4/3
• csc θ = 5/4
• sec θ = 5/3
• cot θ = 3/4

Example Problem 2

Given

• Adjacent = 8
• Hypotenuse = 17
• Find Missing Side: 15 (using Pythagorean theorem)

Calculate Trigonometric Functions:

• sin θ = 15/17
• cos θ = 8/17
• tan θ = 15/8
• csc θ = 17/15
• sec θ = 17/8
• cot θ = 8/15

Example Problem 3

Given:

• Hypotenuse = 25
• Side = 15
• Find Missing Side: 20

Calculate Trigonometric Functions:

• sin θ = 20/25 = 4/5
• cos θ = 15/25 = 3/5
• tan θ = 20/15 = 4/3
• csc θ = 5/4
• sec θ = 5/3
• cot θ = 3/4

Finding Missing Side Using Angle

Example:

• Given θ = 38°, Adjacent = 42
• tan(θ) = x / 42
• x = 42 * tan(38°)
• Result: x ≈ 32.8*

Finding Missing Side (Using Hypotenuse)

Given:

• Hypotenuse = 26, θ = 54°
• cos(θ) = x / 26
• x = 26 * cos(54°)
• Result: x ≈ 15.28*

Finding Missing Angle

Example:

• Opposite = 5, Adjacent = 4
• tan(θ) = 5/4
• θ = arctan(5/4)
• Result: θ ≈ 51.34°

Resources

Trigonometry Course on Udemy:

• Search for "Trigonometry, the Unit Circle, Angles, and Right Triangles"
• Topics include:
  • Angles & Radians
  • Unit Circle
  • Right Triangle Trigonometry
  • Inverse Trigonometric Functions
  • Trigonometric Identities & Equations
  • Law of Sines & Cosines (Future Topics)

These notes summarize the key points covered in the lecture regarding right triangle trigonometry, the use of special triangles, how to calculate trigonometric functions, and resources for further study.