Use the trigonometric substitution to write the algebraic expression as a trigonometric function...

Question:

Use the trigonometric substitution to write the algebraic expression as a trigonometric function of 0, where 0 < θ < π/2.

{eq}\sqrt{x^2+100}, \quad x = 10 \tan \theta {/eq}

Basic Trigonometric Identities:

In trigonometry, we have several basic identities that help us to solve various problems related to six trigonometric functions. Out of these basic identities, Pythagorean identities are the three identities that are based on the Pythagoras theorem and play an important role while solving trigonometric problems.

$$\begin{align} \sin^2x+\cos^2 x &=1\\[0.3cm] 1+\tan^2 x &=\sec^2 x\\[0.3cm] 1+\cot^2 x &=\csc^2 x \end{align} $$

Answer and Explanation: 1


Given data:

$$\sqrt{x^2+100}, \quad x = 10 \tan \theta $$

Using the given trigonometric substitution to write the algebraic expression as a trigonometric function of {eq}\theta {/eq}

The trigonometry indentity that we will apply in the given expression is:

$$1+\tan^2 \theta=\sec^2 \theta $$

Substitute the given trigonometric substitution in the given expression:

$$\begin{align} \sqrt{x^2+100} &=\sqrt{(10\tan \theta)^2+100} \ && \left [ \text{Given}: x = 10 \tan \theta \right ]\\[0.3cm] &=\sqrt{100\tan^2 \theta+100}\\[0.3cm] &=\sqrt{100(1+\tan^2 \theta)}\\[0.3cm] &=\sqrt{100\sec^2 \theta} \ && \left [ \text{Identity}: 1+\tan^2 \theta =\sec^2 \theta \right ]\\[0.3cm] &=\sqrt{(10\sec \theta)^2}\\[0.3cm] &=\boxed{10\sec \theta} \end{align} $$


Learn more about this topic:

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What is Trigonometry? - Functions, Formulas & Applications

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Chapter 22 / Lesson 11
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Learn what trigonometry is and what trigonometric functions are. Understand the examples of how to use each function, as well as know the instances when it is useful to use trigonometry.


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