Lecture on Right Triangle Trigonometry and SOH CAH TOA
Introduction to SOH CAH TOA
Expression used in trigonometry for right triangles.
SOH: Sine = Opposite/Hypotenuse
CAH: Cosine = Adjacent/Hypotenuse
TOA: Tangent = Opposite/Adjacent
Right Triangle Basics
Three sides:
Opposite: Opposite to the angle theta.
Adjacent: Next to the angle theta.
Hypotenuse: Longest side, opposite the right angle.
Pythagorean theorem: (a^2 + b^2 = c^2)
Trigonometric Functions
Sine (sin θ): Opposite/Hypotenuse
Cosine (cos θ): Adjacent/Hypotenuse
Tangent (tan θ): Opposite/Adjacent
Cosecant (csc θ): Reciprocal of sine = Hypotenuse/Opposite
Secant (sec θ): Reciprocal of cosine = Hypotenuse/Adjacent
Cotangent (cot θ): Reciprocal of tangent = Adjacent/Opposite
Special Right Triangles
Examples:
3-4-5, 5-12-13, 8-15-17, 7-24-25 triangles.
Multiples of these ratios are also valid (e.g., 6-8-10, 9-12-15).
Solving Right Triangles
Example 1:
Given sides: 3 and 4; find hypotenuse using the Pythagorean theorem: Hypotenuse = 5.
Trig functions for angle θ:
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 2:
Given sides: 8 and 17; missing side = 15 (using Pythagorean theorem).
Trig 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
Finding Missing Sides Using Trigonometry
Use tangent for opposite/adjacent relations.
Example with angle 38°:
x = 42 * tan(38°) = 32.8
Example with hypotenuse 26 and angle 54°:
x = 26 * cos(54°) = 15.28
Finding Angles Using Inverse Trigonometric Functions
For given sides, use inverse functions:
Example: tan θ = 5/4 → θ = arctan(5/4) = 51.34°
Example: cos θ = 3/7 → θ = arccos(3/7) = 64.62°
Trigonometry Course Overview
Available on Udemy.
Covers topics such as angles, unit circle, right triangle trigonometry, graphing, inverse functions, applications, and advanced topics like verifying identities and solving equations.
Future sections to include laws of sines and cosines, polar coordinates.