Introduction
Before circles, there were triangles. Understanding the 6 fundamental ratios that govern clear geometric relationships.
Past Knowledge
This follows from Similar Triangles (Geometry). If angles match, side ratios match.
Today's Goal
We formalize these ratios as Sine, Cosine, and Tangent (and their reciprocals). The famous mnemonic is SOH CAH TOA.
Future Success
In Calculus (and physics), we often decompose forces or movement into horizontal and vertical components. extracts the horizontal part; extracts the vertical part.
Key Concepts
SOH CAH TOA
For a right triangle with angle :
Sine = Opposite / Hypotenuse
Cosine = Adjacent / Hypotenuse
Tangent = Opposite / Adjacent
Reciprocal Functions
Tip: "S" goes with "C" (Sine/Cosecant, Cosine/Secant).
Worked Examples
Example 1: Finding Ratios
Given a right triangle with legs 3 and 4, find where is opposite the side of length 3.
Find Hypotenuse
Use Pythagorean Theorem: .
.
Apply SOH
Sine = Opposite / Hypotenuse.
Example 2: Solving for a Side
A ladder leans against a wall making a angle with the ground. If the ladder is 12 ft long, how high up the wall does it reach?
Identify Sides
Angle: .
Hypotenuse: 12 (the ladder).
We want the Height (Opposite to the angle).
Choose Ratio
Relative to , we have Opp and Hyp. That is SOH.
.
Solve
.
Example 3: Inverse Trig
Find if in a right triangle.
Interpret Tangent
. If the ratio is 1, then Opposite = Adjacent.
Identify Triangle
A right triangle with equal legs is an Isosceles Right Triangle.
Common Pitfalls
Relative Perspectives
"Opposite" and "Adjacent" depend on which angle you are standing at. The side "Opposite" to angle A is "Adjacent" to angle B. Always double check which angle is .
Real-World Application
GPS and Surveying
Modern surveying (and GPS triangulation) relies entirely on these ratios. If you know the distance to a satellite (hypotenuse) and the angle of elevation, you can calculate the precise altitude coordinates (Opposite side) using Sine.
Practice Quiz
Practice Quiz
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