Questions: Solve the following equation on the interval (0 leq theta<2 pi). [ 4 tan ^2(theta)+8=20 ] A) (0, pi) B) (fracpi3, frac2 pi3, frac4 pi3, frac5 pi3) C) (frac5 pi6, frac11 pi6) D) (fracpi2, fracpi3, fracpi4, fracpi6)

Solution Steps

To solve the equation \(4 \tan^2(\theta) + 8 = 20\) on the interval \(0 \leq \theta < 2\pi\), we first isolate \(\tan^2(\theta)\) by subtracting 8 from both sides and then dividing by 4. This gives us \(\tan^2(\theta) = 3\). Next, we take the square root of both sides to find \(\tan(\theta) = \pm\sqrt{3}\). We then determine the angles \(\theta\) within the given interval that satisfy these conditions.

Starting with the equation: \[ 4 \tan^2(\theta) + 8 = 20 \] we isolate \(\tan^2(\theta)\) by subtracting 8 from both sides: \[ 4 \tan^2(\theta) = 12 \] Next, we divide by 4: \[ \tan^2(\theta) = 3 \]

Taking the square root of both sides gives us: \[ \tan(\theta) = \sqrt{3} \quad \text{or} \quad \tan(\theta) = -\sqrt{3} \]

The angles that satisfy \(\tan(\theta) = \sqrt{3}\) are: \[ \theta = \frac{\pi}{3} + k\pi \quad (k \in \mathbb{Z}) \] Within the interval \(0 \leq \theta < 2\pi\), this gives: \[ \theta = \frac{\pi}{3}, \quad \frac{4\pi}{3} \]

The angles that satisfy \(\tan(\theta) = -\sqrt{3}\) are: \[ \theta = \frac{2\pi}{3} + k\pi \quad (k \in \mathbb{Z}) \] Within the interval \(0 \leq \theta < 2\pi\), this gives: \[ \theta = \frac{2\pi}{3}, \quad \frac{5\pi}{3} \]

Final Answer