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

Problem

sec(θ)=2,θ∈Quadrant I

Solution

1. Identify the given trigonometric ratio and the quadrant. Since sec(θ)=2 and sec(θ)=1/cos(θ) we can determine the cosine value.

cos(θ)=1/2

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

sin2(θ)+(1/2)2=1

3. Solve for sin(θ) Since θ is in Quadrant I, the sine value must be positive.

sin2(θ)=1−1/4

sin2(θ)=3/4

sin(θ)=√(,3)/2

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

tan(θ)=√(,3)/2/1/2

tan(θ)=√(,3)

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

csc(θ)=1/sin(θ)

csc(θ)=2/√(,3)

csc(θ)=(2√(,3))/3

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

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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