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 = arctan 0

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 the square root of three divided by 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 are asked to do this without using a calculator, we will be referencing a unit circle if you do not have one, consider pausing and finding one next. Before we do anything else, I need to consider the domain of the arc tangent which is negative infinity to positive infinity in the range which is between negative pi divided by two and pi divided by two. Since the value we're looking at falls in our domain, we will consider the range negative pi divided by two to pi divided by two. So this range tells us that we are in either quadrant one or quadrant four. Since this tangent value is positive, we must be looking in quadrant one because tangent is positive in quadrant one but negative in quadrant four. Also recall that our unit circle doesn't show us tangent directly. So we need to consider that the tangent of X would be the sine of X divided by the cosine of X. So looking in quadrant one on my unit circle, considering that the square root of three divided by three looks like it has been rationalized. I'm gonna consider what sine divided by cosine would give me this. And I actually think it is at the angle pi divided by six. So I'm going to put it down here just because that's what I'm considering because this would be the sign one half divided by the cosine the square to three divided by three. Oh I'm sorry, squared of three divided by two, which could reduce to one divided by the square to three which when rationalized by multiplying numerator and denominator by the square of three would give us back the square to three divided by three. Since that works, this means pi divided by six is our correct angle and that is answer choice. A so Y equals pi divided by six. Have a nice day.

Find the exact value of each real number y if it exists. Do not use a calculator.
y = arctan 0

Key Concepts

Definition of the Arctangent Function

Tangent Function and Its Values

Exact Values Without a Calculator

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