Find the exact value of each real number y if it exists. Do no...

Question

Find the exact value of each real number y if it exists. Do not use a calculator.

y = tan⁻¹ 1

Accepted Answer

Hello, everyone. We are asked to determine the exact value of the given inverse trigonometric function. Without using a calculator, we are given Y equals the arc tangent of radical three. Our answer choices are A Y equals pi divided by six by equals pi divided by three cy equals pi divided by four dy equals zero. First, since we aren't using a calculator, we want to use a unit circle. So if you do not have one, go find one next recall that our unit circle does not actually give us tangent. So we need to remember that the tangent of X would equal the sine of X divided by the cosine of X. From there. I'm going to consider the domain and range of the arc tangent. So the domain is from negative infinity to positive infinity. Our range is from negative pi divided by two to pi divided by two. Since the square root of three falls in our domain, we are going to work within our range of negative pi divided by two to pi divided by two. This means we are considering quadrants one and four. So it's quadrant one or quadrant four that this falls into since tangent here is positive and it is positive in quadrant one, but not quadrant four, we'll be looking in quadrant one. So if I want this to be the square root of three, I need to find the sine of X divided by the cosine of X, that'll equal the square root of three. So if I want the square root to be in my numerator, I'm gonna consider the angle pi divided by three because in that case, I would have the sine of X be the square root of three divided by two and the cosine of XB one half which does reduce down to the square to three. So in this case, Y equals pi divided by three. And that's answer choice B have a nice day.

Find the exact value of each real number y if it exists. Do not use a calculator.
y = tan⁻¹ 1

Key Concepts

Inverse Tangent Function (arctan)

Tangent Values of Special Angles

Range and Principal Values of Inverse Trigonometric Functions

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