Martin simplified the composed function $$ f(x)=\sin (\arctan x) $$. His work is shown below. Step 1: $$ f(x)=\sin A $$, where $$ A=\arctan x $$ and $$ x= an A $$ Step 2: $$ an A=\frac{ ext { opposite }}{ ext { adjacent }}=\frac{x}{1} $$ Step 3: hypotenuse $$ =\sqrt{1^{2}-x^{2}}=\sqrt{1-x^{2}} $$ Step 4: $$ \sin A=\frac{ ext { opposite }}{ ext { hypotenuse }}=\frac{x}{\sqrt{1-x^{2}}} $$ In which step did Martin make his first error? (1 point) step 1 step 2 step 3 step 4 He did not make an error.

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Martin simplified the composed function $$ f(x)=\sin (\arctan x) $$. His work is shown below. Step 1: $$ f(x)=\sin A $$, where $$ A=\arctan x $$ and $$ x=	an A $$ Step 2: $$ 	an A=\frac{	ext { opposite }}{	ext { adjacent }}=\frac{x}{1} $$ Step 3: hypotenuse $$ =\sqrt{1^{2}-x^{2}}=\sqrt{1-x^{2}} $$ Step 4: $$ \sin A=\frac{	ext { opposite }}{	ext { hypotenuse }}=\frac{x}{\sqrt{1-x^{2}}} $$ In which step did Martin make his first error? (1 point) step 1 step 2 step 3 step 4 He did not make an error.

UpStudy AI · Accepted Answer

Let's review Martin’s work:

1. In Step 1, Martin correctly writes: f(x) = sin(arctan x), letting A = arctan x, so that x = tan A.

2. In Step 2, he correctly sets up a right triangle with opposite side = x and adjacent side = 1 since tan A = x/1.

3. In Step 3, he calculates the hypotenuse. However, using the Pythagorean theorem, the hypotenuse should be √(1² + x²) = √(1 + x²), not √(1 - x²).

4. Step 4 then uses the incorrect hypotenuse, yielding an incorrect expression.

Thus, the first error is in Step 3.
The first error is in Step 3.

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