Find the Other Trig Values in Quadrant III tan(theta) = square root of 3

Problem

tan(θ)=√(,3),θ∈Quadrant III

Solution

1. Identify the given information and the quadrant. In Quadrant III, both the xcoordinate (cosine) and the ycoordinate (sine) are negative.

2. Determine the reference angle. Since tan(θ)=√(,3) the reference angle (θ_ref) is 60 or π/3 radians.

3. Calculate the sine value. In Quadrant III, sin(θ) is negative.

sin(θ)=−√(,3)/2

4. Calculate the cosine value. In Quadrant III, cos(θ) is negative.

cos(θ)=−1/2

5. Find the cosecant value by taking the reciprocal of the sine.

csc(θ)=1/sin(θ)=−2/√(,3)=−(2√(,3))/3

6. Find the secant value by taking the reciprocal of the cosine.

sec(θ)=1/cos(θ)=−2

7. Find the cotangent value by taking the reciprocal of the tangent.

cot(θ)=1/tan(θ)=1/√(,3)=√(,3)/3

Final Answer

sin(θ)=−√(,3)/2,cos(θ)=−1/2,csc(θ)=−(2√(,3))/3,sec(θ)=−2,cot(θ)=√(,3)/3

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