Use the trigonometric substitution $x = 8 \sec ( \theta )$ to write the expression $\sqrt { x ^ { 2 } - 64 }$ as a trigonometric function of $\theta$, where $0

Question

Use the trigonometric substitution $x = 8 \sec ( \theta )$ to write the expression $\sqrt { x ^ { 2 } - 64 }$ as a trigonometric function of $\theta$, where $0 < \theta < \frac { \pi } { 2 }$.
A) $8 \tan ( \theta )$
B) $64 \tan ( \theta )$
C) $64 \sec ( \theta )$
D) $8 \sec ( \theta )$
E) $8 \sec ( \theta ) - 1$

Quizplus · Accepted Answer

Using the trigonometric identity $\sec^2(\theta) - 1 = \tan^2(\theta)$, we substitute $x = 8\sec(\theta)$ into the expression $\sqrt{x^2 - 64}$ to get $\sqrt{(8\sec(\theta))^2 - 64} = \sqrt{64\sec^2(\theta) - 64} = \sqrt{64(\sec^2(\theta) - 1)} = \sqrt{64\tan^2(\theta)} = 8\tan(\theta)$.

Use the Trigonometric Substitution x=8sec⁡(θ)x = 8 \sec ( \theta )x=8sec(θ)

Use the Trigonometric Substitution $x = 8 \sec ( \theta )$