Write each trigonometric expression as an algebraic expression...

Question

Write each trigonometric expression as an algebraic expression in u, for u > 0.

tan (arcsec (√1―u²) / u)

Accepted Answer

Hello, everyone. We are asked to transform the following expression into an algebraic expression in X. We want to assume that the inverse trigonometric function is defined for its arguments. And assume that X is greater than zero. We are given the tangent of the inverse C can of the square root of four minus X squared divided by X. Our answer choices are a, the square root of four minus X squared divided by XB, the square root of four minus two X squared divided by XC, the square root of four plus X squared divided by XD, the square root of four plus two X squared divided by X. First, I'm gonna rewrite our expression. So we have the tangent of the inverse C can of the square root of four minus X squared divided by X. So we're gonna take this apart a little bit first, I'm gonna take what's in our parentheses. So the inverse C can't of the square root of four minus X squared divided by X. And I'm gonna set it to an unknown theta. This way I can rewrite this as the C can of theta equals the square root of four minus X squared divided by X from there using the reciprocal identities. I could rewrite this as the cosine of theta and I invert or take the reciprocal of the fraction. So I'll have X divided by the square root of four minus X squared. And I'm gonna go back to a little bit of right angle trig here. So if you recall cosign was the adjacent side divided by the hypotenuse when we're looking at a right triangle. So if I draw myself a little right triangle, put my right angle here. I know that the sign adjacent to theta which I'm gonna just put in this bottom right corner is X. The hypotenuse is the square root of four minus X squared. And I'm going to call the site opposite the A. So now if I want to use the Pythagorean theorem, I recall that, that states that A squared plus B squared equals C squared where C is the hypotenuse. So we'd have a squared plus X squared equals the square root of four minus X squared squared. So we get a squared plus X squared equals four minus X squared. I want to find the value for A, so I can use it to find the tangent here. So I'm gonna subtract X squared from both sides. So I get a squared equals and I have four minus two X squared. I take the square root of both sides, nugget A equals the square root of four minus two X squared. So I'm gonna write that on my right triangle, that this side is equal to the square root of four minus two X squared. And then if I want to take the tangent of theta, which we replaced theta in our parentheses, I know that again with right triangle trig, this is the opposite side divided by the adjacent side. So opposite theta is the square root of four minus two X squared adjacent to theta is X. So this means written in terms of X, this expression would be the square root of four minus two X squared divided by X. And that is answer choice B have a nice day.

Write each trigonometric expression as an algebraic expression in u, for u > 0.
tan (arcsec (√1―u²) / u)

Key Concepts

Arcsecant Function

Tangent Function

Trigonometric Identities

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