Find the Other Trig Values in Quadrant I sin(theta)=1/6

Problem

sin(θ)=1/6,θ∈Quadrant I

Solution

1. Identify the given information and the quadrant. Since θ is in Quadrant I, all trigonometric ratios (sin() cos() tan() csc() sec() cot() will be positive.

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

(1/6)2+cos2(θ)=1

1/36+cos2(θ)=1

cos2(θ)=1−1/36

cos2(θ)=35/36

3. Solve for cosine by taking the square root. Since θ is in Quadrant I, we take the positive root.

cos(θ)=√(,35/36)

cos(θ)=√(,35)/6

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

tan(θ)=(1/6)/(√(,35)/6)

tan(θ)=1/√(,35)

tan(θ)=√(,35)/35

5. Find the reciprocal identities for cosecant, secant, and cotangent.

csc(θ)=1/sin(θ)=6

sec(θ)=1/cos(θ)=6/√(,35)

sec(θ)=(6√(,35))/35

cot(θ)=1/tan(θ)=√(,35)

Final Answer

cos(θ)=√(,35)/6,tan(θ)=√(,35)/35,csc(θ)=6,sec(θ)=(6√(,35))/35,cot(θ)=√(,35)

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