Find the Other Trig Values in Quadrant I csc(theta)=2

Problem

csc(θ)=2,θ∈Quadrant I

Solution

1. Identify the given trigonometric ratio and the quadrant. Since csc(θ)=2 and θ is in Quadrant I, all trigonometric values will be positive.

2. Find the sine value using the reciprocal identity sin(θ)=1/csc(θ)

sin(θ)=1/2

3. Determine the cosine value using the Pythagorean identity cos2(θ)+sin2(θ)=1

cos2(θ)+(1/2)2=1

cos2(θ)+1/4=1

cos2(θ)=3/4

cos(θ)=√(,3/4)

cos(θ)=√(,3)/2

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

tan(θ)=(1/2)/(√(,3)/2)

tan(θ)=1/√(,3)

tan(θ)=√(,3)/3

5. Find the secant value using the reciprocal identity sec(θ)=1/cos(θ)

sec(θ)=2/√(,3)

sec(θ)=(2√(,3))/3

6. Find the cotangent value using the reciprocal identity cot(θ)=1/tan(θ)

cot(θ)=√(,3)

Final Answer

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

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