Overview
This lecture explains how to solve basic trigonometric equations with one trigonometric function, using algebraic manipulation, trigonometric tables, and the unit circle to find all solutions within a given interval.
Steps to Solve Basic Trigonometric Equations
- Begin by isolating the trigonometric function (e.g., sin(x), cos(x), etc.) on one side of the equation.
- Use inverse operations: addition/subtraction and multiplication/division as needed.
- Once isolated, set the trigonometric function equal to a numerical value.
Using Angle Tables and Unit Circle
- Refer to the table of notable (special) angles to find which angle produces the required value for the trigonometric function.
- Recall that sine and cosine values may have two solutions within 0° to 360° (or 0 to 2π radians) due to the symmetry of the unit circle.
Example 1: Solving 2sin(x) + 3 = 4
- Isolate: 2sin(x) = 1 ⇒ sin(x) = 1/2.
- From the table, sin(30°) = 1/2 ⇒ first solution: x = 30° (or π/6 radians).
- Sine is positive in the first and second quadrants.
- Second solution: x = 180° − 30° = 150° (or 5π/6 radians).
- Verify solutions by substituting back into the original equation.
Example 2: Solving 2cos(x) = √2
- Isolate: cos(x) = √2/2.
- From the table, cos(45°) = √2/2 ⇒ first solution: x = 45° (or π/4 radians).
- Cosine is positive in the first and fourth quadrants.
- Second solution: x = 360° − 45° = 315° (or 7π/4 radians).
- Verify by substitution into the original equation.
Example 3: Solving 4sin(θ) + 5 = 7
- Isolate: 4sin(θ) = 2 ⇒ sin(θ) = 1/2.
- From the table, sin(30°) = 1/2 ⇒ first solution: θ = 30° (or π/6 radians).
- Sine is positive in first and second quadrants.
- Second solution: θ = 180° − 30° = 150° (or 5π/6 radians).
- Verify by substitution.
Key Terms & Definitions
- Trigonometric equation — An equation involving a trigonometric function of a variable.
- Isolate — Manipulate the equation to get the trigonometric function alone on one side.
- Unit circle — A circle of radius 1 centered at the origin, used to determine the sign and values of trig functions in different quadrants.
- Quadrant — Each of the four sections of the coordinate plane, affecting the sign of trig functions.
Action Items / Next Steps
- Practice solving trigonometric equations by isolating the function and using angle tables.
- Complete the two practice equations given at the end of the lecture.
- Review the previous introductory video on trigonometric equations for foundational concepts.