Math Snap

STEP 1

1. The equation $8 \tan(4\theta) + 3 = 11$ is trigonometric.
2. We are solving for $\theta$ in the first quadrant, where $0 < \theta < \frac{\pi}{2}$.
3. We will convert the solution from radians to degrees.

STEP 2

1. Isolate the trigonometric function $\tan(4\theta)$.
2. Solve for $4\theta$.
3. Solve for $\theta$.
4. Convert $\theta$ from radians to degrees.

STEP 3

Subtract 3 from both sides of the equation to isolate the term with the tangent function:
$$8 \tan(4\theta) + 3 = 11$$ $$8 \tan(4\theta) = 11 - 3$$ $$8 \tan(4\theta) = 8$$

STEP 4

Divide both sides by 8 to solve for $\tan(4\theta)$:
$$\tan(4\theta) = \frac{8}{8}$$ $$\tan(4\theta) = 1$$

STEP 5

Find $4\theta$ by taking the arctangent of both sides. Since $\tan(\frac{\pi}{4}) = 1$, we have:
$$4\theta = \frac{\pi}{4}$$ Solve for $\theta$ by dividing both sides by 4:
$$\theta = \frac{\pi}{4} \times \frac{1}{4}$$ $$\theta = \frac{\pi}{16}$$

Convert $\theta$ from radians to degrees. We know that $180^\circ = \pi$ radians, so:
$$\theta = \frac{\pi}{16} \times \frac{180^\circ}{\pi}$$ $$\theta = \frac{180^\circ}{16}$$ $$\theta = 11.25^\circ$$ The value of $\theta$ is:
$$\theta = \frac{\pi}{16} \text{ radians, or } 11.25^\circ$$