Introduction
When you solve , you might think . But wait—sine is also at ! And due to periodicity, there are infinitely many solutions.
Prerequisite Connection
You know the unit circle, reference angles, and how to evaluate trig functions.
Today's Increment
We're solving linear trig equations and finding all solutions in a given interval or the general solution.
Why This Matters
In Physics and Engineering, periodic phenomena (waves, oscillations) require solving trig equations to find critical points.
Solving Strategy
Isolate the Trig Function
Get the equation in the form or .
Find Reference Angle
Use inverse trig or unit circle knowledge to find the basic angle.
Identify All Quadrants
Determine which quadrants give the correct sign for the trig value.
Write General Solution (if needed)
Add (or ) where is any integer.
Worked Examples
Example 1: Basic Equation
Solve on .
Find Reference Angle
Identify Quadrants
Sine is positive in QI and QII.
Find All Solutions
QI:
QII:
Answer:
Example 2: General Solution
Find all solutions to .
Identify Where
This happens at on the unit circle.
Add Period
Cosine repeats every .
Answer: , where
Example 3: Factoring (Advanced)
Solve on .
Do NOT Divide by !
Move everything to one side:
Factor
Solve Each Factor
Answer:
Common Pitfalls
Dividing by a trig function
If you divide by , you lose solutions where . Always factor instead.
Missing a quadrant
Remember ASTC: All, Sine, Tangent, Cosine are positive in QI, QII, QIII, QIV respectively.
Real-World Application
Tide Predictions
Ocean tides follow sinusoidal patterns. Coastal engineers solve trig equations to predict when high and low tides will occur, which is critical for shipping schedules, coastal construction, and flood planning.
Practice Quiz
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