Find the Tangent Given the Point (( square root of 3)/2,-1/2)

Problem

(√(,3)/2,−1/2)

Solution

1. Identify the coordinates of the given point on the unit circle as x=√(,3)/2 and y=−1/2

2. Recall the trigonometric definition of the tangent function in terms of coordinates on the unit circle, which is tan(θ)=y/x

3. Substitute the given values into the ratio.

tan(θ)=(−1/2)/√(,3)/2

4. Simplify the complex fraction by multiplying the numerator by the reciprocal of the denominator.

tan(θ)=−1/2⋅2/√(,3)

5. Cancel the common factor of 2 from the numerator and denominator.

tan(θ)=−1/√(,3)

6. Rationalize the denominator by multiplying the numerator and denominator by √(,3)

tan(θ)=−(1⋅√(,3))/(√(,3)⋅√(,3))

tan(θ)=−√(,3)/3

Final Answer

tan(θ)=−√(,3)/3

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