Find the Other Trig Values in Quadrant III sec(theta)=-2

Problem

sec(θ)=−2,θ∈Quadrant III

Solution

1. Identify the given trigonometric ratio and the quadrant. Since sec(θ)=−2 the reciprocal identity cos(θ)=1/sec(θ) gives cos(θ)=−1/2 In Quadrant III, both sine and cosine are negative, while tangent and cotangent are positive.

2. Determine the value of sin(θ) using the Pythagorean identity sin2(θ)+cos2(θ)=1

sin2(θ)+(−1/2)2=1

sin2(θ)+1/4=1

sin2(θ)=3/4

3. Solve for sin(θ) by taking the square root. Since θ is in Quadrant III, the sine value must be negative.

sin(θ)=−√(,3/4)

sin(θ)=−√(,3)/2

4. Calculate csc(θ) using the reciprocal identity csc(θ)=1/sin(θ)

csc(θ)=1/(−√(,3)/2)

csc(θ)=−2/√(,3)

csc(θ)=−(2√(,3))/3

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

tan(θ)=(−√(,3)/2)/(−1/2)

tan(θ)=√(,3)

6. Calculate cot(θ) using the reciprocal identity cot(θ)=1/tan(θ)

cot(θ)=1/√(,3)

cot(θ)=√(,3)/3

Final Answer

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

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