Find the Other Trig Values in Quadrant I sec(theta)=9

Problem

sec(θ)=9,θ∈Quadrant I

Solution

1. Identify the relationship between the secant function and the cosine function.

cos(θ)=1/sec(θ)

2. Substitute the given value of sec(θ) to find cos(θ)

cos(θ)=1/9

3. Apply the Pythagorean identity sin2(θ)+cos2(θ)=1 to solve for sin(θ)

sin2(θ)+(1/9)2=1

4. Simplify the equation and solve for sin(θ) noting that sin(θ) is positive in Quadrant I.

sin2(θ)=1−1/81

sin2(θ)=80/81

sin(θ)=√(,80/81)

sin(θ)=(4√(,5))/9

5. Calculate tan(θ) using the identity tan(θ)=sin(θ)/cos(θ)

tan(θ)=(4√(,5))/9/1/9

tan(θ)=4√(,5)

6. Calculate cot(θ) using the identity cot(θ)=1/tan(θ)

cot(θ)=1/(4√(,5))

cot(θ)=√(,5)/20

7. Calculate csc(θ) using the identity csc(θ)=1/sin(θ)

csc(θ)=1/(4√(,5))/9

csc(θ)=9/(4√(,5))

csc(θ)=(9√(,5))/20

Final Answer

sin(θ)=(4√(,5))/9,cos(θ)=1/9,tan(θ)=4√(,5),csc(θ)=(9√(,5))/20,cot(θ)=√(,5)/20

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