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Chapter 4: Problem 66

Write an algebraic expression that is equivalent to the given expression. $$\sin (\arctan x)$$

Short Answer

Expert verified
The algebraic expression equivalent to \( \sin (\arctan x) \) is \( \frac{x}{\sqrt{x^{2} + 1}} \).

Step by step solution

01

Set up a right triangle

Start by setting up a right triangle where \( \arctan x \) is one of the angles. Taking \( x \) as a ratio, let us assign the opposite side as \( x \) and the adjacent side as \( 1 \). The hypotenuse \( h \) of the triangle can then be calculated using the Pythagorean theorem.
02

Apply Pythagorean theorem

By the Pythagorean theorem, the hypotenuse \( h \) is given by \( h = \sqrt{x^{2} + 1} \) .
03

Find equivalent expression

Now, recall the trigonometric identity for sine, which is given by \( \sin (\theta) = \frac{opposite}{hypotenuse} \). Here, \( \theta = \arctan x \). So, in our triangle, \( \sin (\arctan x) = \frac{x}{\sqrt{x^{2} + 1}} \).

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