Overview
This lecture explains how to solve basic trigonometric equations with a single trigonometric function and demonstrates step-by-step solutions for both sine and cosine equations, including reasoning using the unit circle.
Solving Basic Trigonometric Equations
- Trigonometric equations often require isolating a single trigonometric function (e.g., sin(x), cos(x), tan(x)).
- Rearranging terms allows you to solve for the trigonometric function (e.g., 2sin(x) + 3 = 4 becomes sin(x) = 1/2).
- Use known values from the table of notable angles or the unit circle when no calculator is available.
- The main solution corresponds to a known angle (e.g., sin(x) = 1/2 means x = 30° or π/6 radians).
- Trigonometric equations can have multiple solutions within a given interval (e.g., [0°, 360°]).
Finding All Solutions in a Given Interval
- Use the unit circle to identify all quadrants where the function has the required sign (positive or negative).
- For sin(x) = 1/2, sine is positive in the 1st and 2nd quadrants: solutions are x₁ = 30°, x₂ = 150° (or π/6, 5π/6).
- For cos(x) = √2/2, cosine is positive in the 1st and 4th quadrants: solutions are x₁ = 45°, x₂ = 315° (or π/4, 7π/4).
- For each additional solution, subtract or add the reference angle from 180° or 360° accordingly.
Verifying Solutions
- Substitute solutions back into the original equation to confirm correctness.
- Use a calculator where possible for verification.
Practice Examples
- Example: 2cos(x) = √2 solves to x = 45°, 315°.
- Example: 4sin(θ) + 5 = 7 solves to θ = 30°, 150°.
Key Terms & Definitions
- Trigonometric Equation — An equation involving trigonometric functions of a variable.
- Notable Angles — Common angles (like 30°, 45°, 60°) with known trigonometric values.
- Unit Circle — A circle with radius 1 used to determine signs and values of trig functions in different quadrants.
- Reference Angle — The acute angle a terminal side makes with the x-axis.
Action Items / Next Steps
- Practice solving: 2cos(x) = √2 and 4sin(θ) + 5 = 7 as shown in the examples.
- Review and memorize the trigonometric values for notable angles.
- Make sure to understand the use of the unit circle in finding additional solutions.