Step by Step
1
Isolate the trig function
Rearrange the equation so the trig function (like sin(x) or cos(x)) is completely alone on one side.
2
Find the reference angle
Use the inverse trig function to find the reference angle — the acute angle version of the solution, ignoring sign for now.
3
Use ASTC/CAST to find all quadrant solutions between 0° and 360°
Based on whether the original value was positive or negative, use ASTC (All Students Take Calculus) to determine which quadrants the actual solutions fall in, then find the corresponding angles in those quadrants.
4
Add the period for the general solution
Since trig functions repeat periodically, add n×360° (or n×2π in radians) to each solution found in step 3 to express the complete general solution — sine and cosine have a period of 360°/2π, while tangent has a period of only 180°/π.
Applied Walkthrough
1
Solve sin(x) = 1/2 for all solutions. First, the equation is already isolated: sin(x) = 1/2.
2
Find the reference angle: arcsin(1/2) = 30°.
3
Since sin(x) is positive, use ASTC: sine is positive in Quadrant I and Quadrant II. In Quadrant I, the solution is 30° itself. In Quadrant II, the solution is 180° − 30° = 150°.
4
Add the period (360°) to express the full general solution: x = 30° + 360°n, or x = 150° + 360°n, where n is any integer.
Exam Application
Exams test whether you can execute all four steps correctly — isolating the function, finding the reference angle, using ASTC/CAST to identify all quadrant solutions, and correctly adding the appropriate period for the general solution.
⚠ Common Trap
The most common trap is finding just one solution (from the reference angle) and stopping there, forgetting that trig equations typically have multiple solutions within one full rotation, plus infinitely more repeating every period.
✓ Quick Self-Check
1. What is the first step in solving a trigonometric equation?
Isolate the trig function on one side of the equation.
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2. What does CAST/ASTC help you determine when solving trig equations?
Which quadrants contain valid solutions, based on the sign of the trig value.
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3. Solve sin(x) = 1/2 for all solutions between 0° and 360°.
x = 30° and x = 150°.
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4. What is the period used for the general solution of sine or cosine equations?
360° (or 2π radians).
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5. What is the period used for the general solution of tangent equations?
180° (or π radians) — half the period of sine and cosine.
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