Find the exact value of each real number y. Do not use a calcu...

Question

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

y = tan⁻¹ (―√3)

Accepted Answer

Hello, today we are going to be determining the exact value of the given inverse trigonometric function without using a calculator. So we are given Y is equal to the tangent inverse of negative square root three divided by three plus the tangent inverse of the square root of three. So before we jump into this problem, we need to understand the domain of the tangent inverse function. The domain of the tangent inverse function is given to us as negative pi divided by two comma pi divided by two. But we do not include the values of negative pi divided by two or pi divided by two because the tangent inverse function is undefined at these values. What this means is that on the unit circle, the domain of tangent inverse is the right half of the unit circle, not including the values of pi divided by two or negative pi divided by two. Now, what we are going to do is we're going to evaluate each of the true inverse trigonometric functions individually. Let's start with tangent inverse of negative square root three divided by three. Now recall that whenever we solve for an inverse trigonometric function, the output is always going to be some angle theta. So what we can do is we can take the tangent inverse of negative square root three divided by three and set it equal to theta. If we take the tangent function of both sides of this equation, we can rewrite the equation to be the negative square root three divided by three equal to tangent of the. And if we reorder the terms in the equation, we can rewrite the equation to be tangent of data is equal to the negative square root three divided by three. So what this means is that we want to figure out or we want to ask ourselves what angle can I plug into tangent? That will give me the output of negative square root three divided by three. Furthermore, this angle must be within the domain of tangent inverse. While tangent data is equal to negative square root three divided by three at the angle negative pi divided by six. So what this means is that the inverse function tangent inverse of negative square root three divided by three will equal to negative pi divided by six. This is going to be the value of the first inverse trigonometric function. Now let's solve for the second inverse trigonometric function just like before we're going to take tangent inverse of the square root of three and set it equal to some angle theta we're going to take the tangent function of both sides of this equation and that will give us the square root of three is equal to tangent theta. And reordering the terms will give us the equation tangent of theta is equal to the square root of three. Now again, we want to ask ourselves, what is an angle theta that we can plug into the tangent function that will give us the output of the square root of three. And is also within the domain from negative square, from negative pi divided by two to pi divided by two. Well tangent theta is equal to the square root of three at the angle pi divided by three. So what this means is that the inverse function tangent inverse of the square root of three will equal to pi divided by three. This is going to be the value of the second inverse trigonometric function. Now that we have the value of both inverse trig functions we can now solve for the inverse trigonometric function. So again, we are given Y is equal to tangent inverse of the negative square root three divided by three plus a tangent inverse of the square root of three. Using the values we just solved, we can rewrite the equation as Y is equal to negative pi divided by six plus pi divided by three, we can in order to add both of these fractions together, we need to have the same denominator. We can multiply the right hand fraction by 2/2. This will allow us to rewrite the equation as Y is equal to negative pi divided by six plus two pi divided by six. And simplifying this will give us a final output of pi divided by six. This is going to be the exact value of the inverse trigonometric function. And with that being said, the answer to this problem is going to be D. So I hope this video helps you in understanding how to approach this problem. And I'll go ahead and see you all in the next video.

Find the exact value of each real number y. Do not use a calculator.
y = tan⁻¹ (―√3)

Key Concepts

Inverse Trigonometric Functions

Tangent Function

Quadrants and Angle Values

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