Find the Other Trig Values in Quadrant I csc(theta)=3

Problem

csc(θ)=3,θ∈Quadrant I

Solution

1. Identify the given trigonometric ratio and the quadrant. Since csc(θ)=3 and csc(θ)=1/sin(θ) we can determine the value of sin(θ)

sin(θ)=1/3

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

(1/3)2+cos2(θ)=1

3. Solve for cos2(θ) by subtracting the squared sine value from both sides.

cos2(θ)=1−1/9

cos2(θ)=8/9

4. Take the square root of both sides. Since θ is in Quadrant I, all trigonometric values are positive.

cos(θ)=√(,8/9)

cos(θ)=(2√(,2))/3

5. Find tan(θ) using the ratio sin(θ)/cos(θ)

tan(θ)=(1/3)/(2√(,2)/3)

tan(θ)=1/(2√(,2))

tan(θ)=√(,2)/4

6. Find cot(θ) by taking the reciprocal of tan(θ)

cot(θ)=2√(,2)

7. Find sec(θ) by taking the reciprocal of cos(θ)

sec(θ)=3/(2√(,2))

sec(θ)=(3√(,2))/4

Final Answer

sin(θ)=1/3,cos(θ)=(2√(,2))/3,tan(θ)=√(,2)/4,sec(θ)=(3√(,2))/4,cot(θ)=2√(,2)

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