Find the Other Trig Values in Quadrant I sec(theta)=6

Problem

sec(θ)=6,θ∈Quadrant I

Solution

1. Identify the given trigonometric ratio and the quadrant. Since sec(θ)=6 and θ is in Quadrant I, all trigonometric functions will be positive.

2. Find the cosine value using the reciprocal identity.

cos(θ)=1/sec(θ)

cos(θ)=1/6

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

sin2(θ)+(1/6)2=1

sin2(θ)+1/36=1

sin2(θ)=35/36

sin(θ)=√(,35/36)

sin(θ)=√(,35)/6

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

tan(θ)=√(,35)/6/1/6

tan(θ)=√(,35)

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

cot(θ)=1/√(,35)

cot(θ)=√(,35)/35

6. Find the cosecant value using the reciprocal identity csc(θ)=1/sin(θ)

csc(θ)=1/√(,35)/6

csc(θ)=6/√(,35)

csc(θ)=(6√(,35))/35

Final Answer

sin(θ)=√(,35)/6,cos(θ)=1/6,tan(θ)=√(,35),csc(θ)=(6√(,35))/35,cot(θ)=√(,35)/35

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