Use the trigonometric substitution to rewrite the algebraic expression as a trigonometric function of θ,where $0

Question

Use the trigonometric substitution to rewrite the algebraic expression as a trigonometric function of θ,where $0 < x < \frac { \pi } { 2 }$ . $\sqrt { x ^ { 2 } + 25 } , x = 5 \tan \theta$ ​
A)- 5 sec θ
B)5 tan θ
C)25 sec θ
D)25 tan θ
E)5 sec θ

Quizplus · Accepted Answer

Using the given substitution $x = 5\tan\theta$, the expression becomes $\sqrt{(5\tan\theta)^2 + 25}$. Simplifying inside the square root gives $\sqrt{25\tan^2\theta + 25}$, which can be factored as $\sqrt{25(\tan^2\theta + 1)}$. Using the trigonometric identity $\tan^2\theta + 1 = \sec^2\theta$, the expression simplifies to $\sqrt{25\sec^2\theta}$, which is $5\sec\theta$.

Use the Trigonometric Substitution to Rewrite the Algebraic Expression as a Trigonometric