Write an algebraic expression that is equivalent to $\tan ( \arccos 3 x )$ .
A) $\frac { \sqrt { 1 - 9 x ^ { 2 } } } { 3 x }$
B) $\sqrt { 1 - 9 x ^ { 2 } }$
C) $\frac { \sqrt { 1 - 9 x ^ { 2 } } } { 1 - 9 x ^ { 2 } }$
D) $\frac { 1 } { 3 x }$
E) $3 x$

Question

Quizplus · Accepted Answer

The expression $\tan ( \arccos 3 x )$ can be solved using trigonometric identities. Let $\theta = \arccos 3x$, then $\cos \theta = 3x$. Since $\tan \theta = \frac{\sin \theta}{\cos \theta}$, and knowing that $\sin^2 \theta + \cos^2 \theta = 1$, we can solve for $\sin \theta = \sqrt{1 - \cos^2 \theta} = \sqrt{1 - (3x)^2}$. Therefore, $\tan \theta = \frac{\sqrt{1 - 9x^2}}{3x}$, making option A correct.

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Write an Algebraic Expression That Is Equivalent To $\tan ( \arccos 3 x )$