Write the following expression in algebraic form. $\tan \left( \operatorname { arcsec } \left( \frac { x } { 8 } \right) \right)$
A) $x ^ { 2 } - 64$
B) $\frac { \sqrt { x ^ { 2 } - 64 } } { 8 }$
C) $1 + 64 x ^ { 2 }$
D) $\sqrt { x ^ { 2 } - 8 }$
E) $1 + 8 x ^ { 2 }$

Question

Quizplus · Accepted Answer

To express $\tan(\operatorname{arcsec}(\frac{x}{8}))$ in algebraic form, we start by considering a right triangle where the secant (the reciprocal of the cosine) is $\frac{x}{8}$. In terms of a triangle, secant is the ratio of the hypotenuse over the adjacent side, so if the adjacent side (next to the angle) is 8, the hypotenuse is $x$. To find the opposite side, we use the Pythagorean theorem: $x^2 = 8^2 + \text{opposite}^2$, which simplifies to $\text{opposite}^2 = x^2 - 64$, and thus the opposite side is $\sqrt{x^2 - 64}$. The tangent of the angle is the ratio of the opposite side over the adjacent side, which gives $\frac{\sqrt{x^2 - 64}}{8}$.

Write the Following Expression in Algebraic Form tan⁡(arcsec⁡(x8))\tan \left( \operatorname { arcsec } \left( \frac { x } { 8 } \right) \right)tan(arcsec(8x​))

Write the Following Expression in Algebraic Form $\tan \left( \operatorname { arcsec } \left( \frac { x } { 8 } \right) \right)$