Solving Trigonometric Equations

Precalculus Honors

Brian Risden

Solving on \([0, 2\pi)\)

Problem: Solve \(2\cos(x) + \sqrt{2} = 0\)

Step 1: Isolate the function \[2\cos(x) = -\sqrt{2}\] \[\cos(x) = \color{red}{-\frac{\sqrt{2}}{2}}\]

Step 2: Quadrant Check Since the ratio is \(\color{red}{\text{negative}}\), \(x\) must be in: \[\text{Quadrant } \color{blue}{II} \text{ or Quadrant } \color{purple}{III}\]

Finding the Values

Step 3: Reference Angle \[\cos(\theta') = \frac{\sqrt{2}}{2} \implies \theta' = \color{orange}{\frac{\pi}{4}}\]

Step 4: Final Solutions \[\underbrace{x = \pi - \frac{\pi}{4}}_{\text{QII}} = \color{blue}{\frac{3\pi}{4}}\] \[\underbrace{x = \pi + \frac{\pi}{4}}_{\text{QIII}} = \color{purple}{\frac{5\pi}{4}}\]

Solution Set: \(S = \left\{ \frac{3\pi}{4}, \frac{5\pi}{4} \right\}\)