Find a solution for the equation. Assume that all angles are acute angles.
-$$\tan \left( 3 \alpha + 12 ^ { \circ } \right) = \cot \left( \alpha + 36 ^ { \circ } \right)$$
A) $10.5 ^ { \circ }$
B) $16 ^ { \circ }$
C) $14.5 ^ { \circ }$
D) $12 ^ { \circ }$

Question

Quizplus · Accepted Answer

The given equation is $\tan(3\alpha + 12^\circ) = \cot(\alpha + 36^\circ)$. We know that $\cot \theta = \tan(90^\circ - \theta)$. Therefore, we can rewrite the equation as $\tan(3\alpha + 12^\circ) = \tan(90^\circ - (\alpha + 36^\circ))$. Simplifying the right side gives us $\tan(3\alpha + 12^\circ) = \tan(54^\circ - \alpha)$. For two angles to have the same tangent value, they must be equal, leading to $3\alpha + 12^\circ = 54^\circ - \alpha$. Solving for $\alpha$ gives $4\alpha = 42^\circ$, so $\alpha = 10.5^\circ$.

Find a Solution for the Equation tan⁡(3α+12∘)=cot⁡(α+36∘)\tan \left( 3 \alpha + 12 ^ { \circ } \right) = \cot \left( \alpha + 36 ^ { \circ } \right)tan(3α+12∘)=cot(α+36∘)

Find a Solution for the Equation $\tan \left( 3 \alpha + 12 ^ { \circ } \right) = \cot \left( \alpha + 36 ^ { \circ } \right)$