Question

Write the composed trigonometric function sin(arctan x) in terms of x. Explain your steps and/or show your work. Remember to rationalize the denominator if necessary .

Answer 1

[AI-generated answer — Bot GPT 3.5]

To write the composed trigonometric function sin(arctan x) in terms of x, we first need to use the definition of the inverse trigonometric function arctan(x).

arctan(x) is defined as the angle whose tangent is x. So, let's consider a right triangle with opposite side = x and adjacent side = 1. Then, the tangent of the angle θ in this triangle is x/1 = x. Using the Pythagorean theorem, the hypotenuse of the triangle is sqrt(1 + x^2).

Now, we can find sin(arctan(x)) using the definition of sine in a right triangle as opposite side / hypotenuse. So, sin(arctan(x)) = x / sqrt(1 + x^2).

Therefore, sin(arctan(x)) = x / sqrt(1 + x^2).

This is the composed trigonometric function sin(arctan x) in terms of x.

Answer 2

can you explain it better

Answer 3

[AI-generated answer — Bot GPT 3.5]

Sure! Let's break it down step by step:

1. We start with the expression sin(arctan(x)). This means we need to find the sine of the angle whose tangent is x.

2. Let's call the angle whose tangent is x as θ. Therefore, tan(θ) = x.

3. Let's consider a right triangle with an angle θ, opposite side x, and adjacent side 1. Using the Pythagorean theorem, the hypotenuse of the triangle is sqrt(1 + x^2).

4. Now, we can find the sine of θ by using the definition of sine in a right triangle: sin(θ) = opposite/hypotenuse.

5. Plugging in the values from our triangle, we get sin(θ) = x / sqrt(1 + x^2).

6. Therefore, sin(arctan(x)) = x / sqrt(1 + x^2).

This is the final expression for the composed trigonometric function sin(arctan(x)) in terms of x.