Match each expression in Column I with its value in Column II....

Question

Match each expression in Column I with its value in Column II.

(2 tan (π/3))/(1 - tan² (π/3))

Accepted Answer

Hello, everyone. We are asked to determine the value of the trigonometric expression. Open parentheses, 2, 10 of seven pi divided by 12 closed parentheses, divided by open parentheses, one minus tangent squared of seven pi divided by 12 closed parentheses. Our answer choices are A zero B, the square root of two divided by two C, the square root of three divided by 3d negative one half. First, I recognize that the expression we're given looks very similar to the tangent double angle formula which is the tangent of two A equals two tangent of A divided by one minus the tangent squared of A. That is what we have here. They have it written horizontally versus vertically, but that's what we're working with. So the angle we're working with are A is the value seven pi divided by 12. So I'm gonna put that into tangent of two A. So the tangent of two multiplied by seven pi divided by 12. And when that's simplified, we will get the tangent of seven pi divided by six. So to find the value from here, I need to reference a unit circle, our unit circles have sine and cosine. So I'm gonna rewrite tangent as the sine of seven pi divided by six, divided by the cosine of seven pi divided by six. So going to my unit circle, I find that the sine of seven pi divided by six is negative one half and the cosine of seven pi divided by six is negative radical three divided by two. So now I'm going to put those in as division. So sine divided by cosine. So we have negative one half divided by negative radical three divided by two. I recall when I have a complex fraction, if both of the fractions have the same denominator, they will cancel out. So this leaves me with negative one divided by negative radical three and a negative divided by a negative is a positive. So I have one divided by radical three. I want to rationalize my denominator by multiplying numerator and denominator by the square root of three. And I get the square root of three divided by three as my answer, this will be the value of the expression we were given. And that is answer choice. C the square root of three divided by three have a nice day.

Match each expression in Column I with its value in Column II.
(2 tan (π/3))/(1 - tan² (π/3))

Key Concepts

Tangent Function and Its Values

Double-Angle Formula for Tangent

Simplification of Trigonometric Expressions

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