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

Problem

sec(θ)=−2,θ∈Quadrant II

Solution

1. Identify the given trigonometric ratio and the quadrant. Since sec(θ)=−2 the cosine value is the reciprocal:

cos(θ)=−1/2

2. Determine the reference angle. In Quadrant II, the cosine is negative. The angle θ whose cosine is −1/2 in the second quadrant is 120 or (2*π)/3 radians.

3. Calculate the sine value using the Pythagorean identity sin2(θ)+cos2(θ)=1 In Quadrant II, sin(θ) is positive:

sin(θ)=√(,1−(−1/2)2)

sin(θ)=√(,1−1/4)

sin(θ)=√(,3)/2

4. Find the cosecant value by taking the reciprocal of the sine:

csc(θ)=1/sin(θ)

csc(θ)=2/√(,3)

csc(θ)=(2√(,3))/3

5. Find the tangent value using the ratio of sine to cosine:

tan(θ)=sin(θ)/cos(θ)

tan(θ)=√(,3)/2/(−1/2)

tan(θ)=−√(,3)

6. Find the cotangent value by taking the reciprocal of the tangent:

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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