In Exercises 83–94, use a right triangle to write each expression...

Question

In Exercises 83–94, use a right triangle to write each expression as an algebraic expression. Assume that x is positive and that the given inverse trigonometric function is defined for the expression in x.

sin (tan⁻¹ x)

Accepted Answer

Hello, everyone. We are asked to transform the following expression into an algebraic expression. We're going to use a right triangle in writing the algebraic expression assume that the inverse trigonometric function is defined for its argument. And we're also going to assume that X is greater than zero. The expression we are given is the sign of the arc tangent of four X. We are given four answer choices all of which are algebraic expressions but differ in if they are fractions or not to begin with, I'm going to draw a right triangle. So it will help me reference where pieces are. And then I'm gonna start in the parentheses of this expression. So I'm gonna start with the arc tangent A four X. I'm gonna set that equal to a angle. I'm calling theta and I'm gonna put that in the bottom right corner of my right triangle. So I have the arc tangent of four X equals the, if I take the tangent of both sides, I get the tangent of theta equals four X. Recalling that when we use a right triangle, I know that tangent is the same as the opposite like divided by the adjacent leg, I can think of this as four X divided by one. So the side, opposite angle theta is four X, the side adjacent to it has a value of one. When we're done, we're gonna need to find the sign of theta. So I'm gonna think about what that ratio is before I do anything else. And I recall the sign of theta is the opposite leg divided by the hypotenuse. We don't know the hypotenuse. I'm gonna call that A and we're going to use the Pythagorean theorem to solve for it. So our legs squared. So four X in parentheses, squared plus one squared has equal A squared, four X squared is gonna be the same as 16 X squared plus one. And that'll equal A squared I want to solve for a. So I'm gonna take the square root of both sides and make sure I take the square root of the entire side and A will equal the square root of 16 X squared plus one. So all of that is under my radical now that I know my hypotenuse, I can go back to finding the sign of the. So my opposite side is four X, my hypotenuse is the square root of 16 X squared plus one. And as much as I wish we were done, we should rationalize this denominator. So we're gonna multiply the numerator and denominator by the radical 16 X squared plus one. And when we do So our new numerator is four X multiplied by the square root of 16 X squared plus one, divided by 16 X squared plus one. So this is our value. We will have it rewritten as an algebraic expression and it will match answer choice. D have a nice day.

In Exercises 83–94, use a right triangle to write each expression as an algebraic expression. Assume that x is positive and that the given inverse trigonometric function is defined for the expression in x. sin (tan⁻¹ x)

Key Concepts

Inverse Trigonometric Functions

Right Triangle Relationships

Pythagorean Identity

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