Find the Other Trig Values in Quadrant III sin(theta)=-1/2

Problem

sin(θ)=−1/2,θ∈Quadrant III

Solution

1. Identify the given information and the signs of trigonometric functions in Quadrant III. In Quadrant III, sin(θ) and cos(θ) are negative, while tan(θ) and cot(θ) are positive.

2. Apply the Pythagorean identity sin2(θ)+cos2(θ)=1 to find the value of cos(θ)

(−1/2)2+cos2(θ)=1

1/4+cos2(θ)=1

cos2(θ)=3/4

3. Solve for cos(θ) by taking the square root, choosing the negative value because θ is in Quadrant III.

cos(θ)=−√(,3/4)

cos(θ)=−√(,3)/2

4. Calculate tan(θ) using the identity tan(θ)=sin(θ)/cos(θ)

tan(θ)=(−1/2)/(−√(,3)/2)

tan(θ)=1/√(,3)

tan(θ)=√(,3)/3

5. Determine the reciprocal functions csc(θ) sec(θ) and cot(θ)

csc(θ)=1/sin(θ)=−2

sec(θ)=1/cos(θ)=−2/√(,3)=−(2√(,3))/3

cot(θ)=1/tan(θ)=√(,3)

Final Answer

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

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