7.3

Solving Using Algebra

• Goal: Solve trigonometric equations using standard algebraic techniques.
• Example: Solve 2 cos^2(\theta) - 1 = 0
  • Solution set on [0, 2\pi): {\frac{\pi}{4}, \frac{7\pi}{4}}
  • General solution: \theta = \frac{\pi}{4} + n\pi, \theta = \frac{7\pi}{4} + n\pi
• To solve equations with multiple trigonometric functions, rearrange and simplify.

Equations of Quadratic Type

• Trigonometric equation of quadratic type looks like a quadratic equation with trigonometric functions.
• Solve by factoring or using the quadratic formula.
• Example: Find all solutions of 2 sin^2(x) - sin(x) - 1 = 0 in [0, 2\pi).
• When squaring both sides of an equation, check for extraneous solutions.

Equations Involving Multiple Angles

• Solve trigonometric equations involving multiple angles by finding solutions in the given interval and then generalizing.
• Example: Solve 2 cos(3t) - 1 = 0. Give solutions in the interval [0,2\pi) and then give a general solution.

Using Inverse Functions

• Use inverse trigonometric functions to solve trigonometric equations.
• Example: Find the general solution of sec^2(x) - 2 tan(x) = 4.
• Example: Find all solutions of sin^2(x) + 6 sin(x) - 2 = 0 in the interval [0, 2\pi).