Write an algebraic expression that is equivalent to $\tan ( \arccos 3 x )$.

A) $\frac { 1 } { 3 x }$

B) $\frac { \sqrt { 1 - 9 x ^ { 2 } } } { 3 x }$

C) $\sqrt { 1 - 9 x ^ { 2 } }$

D) $\frac { 1 } { \sqrt { 1 - 9 x ^ { 2 } } }$

E) $3 x$

Question

Write an algebraic expression that is equivalent to $\tan ( \arccos 3 x )$.

A) $\frac { 1 } { 3 x }$

B) $\frac { \sqrt { 1 - 9 x ^ { 2 } } } { 3 x }$

C) $\sqrt { 1 - 9 x ^ { 2 } }$

D) $\frac { 1 } { \sqrt { 1 - 9 x ^ { 2 } } }$

E) $3 x$

Quizplus · Accepted Answer

To find an expression equivalent to $\tan(\arccos(3x))$, we use the trigonometric identity $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$. Since $\arccos(3x)$ gives an angle whose cosine is $3x$, we can represent this in a right triangle where the adjacent side is $3x$ and the hypotenuse is $1$ (because the cosine of an angle is the adjacent side over the hypotenuse). Using the Pythagorean theorem, the opposite side is $\sqrt{1 - (3x)^2} = \sqrt{1 - 9x^2}$. Therefore, $\tan(\arccos(3x)) = \frac{\sqrt{1 - 9x^2}}{3x}$.

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