Find the Other Trig Values in Quadrant I sin(theta)=1/3

Problem

sin(θ)=1/3,θ∈Quadrant I

Solution

1. Identify the given information and the quadrant. In Quadrant I, all trigonometric functions (sin() cos() tan() csc() sec() cot() are positive.

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

(1/3)2+cos2(θ)=1

3. Solve for cos(θ) by subtracting and taking the square root. Since θ is in Quadrant I, we take the positive root.

1/9+cos2(θ)=1

cos2(θ)=8/9

cos(θ)=√(,8/9)=(2√(,2))/3

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

tan(θ)=(1/3)/(2√(,2)/3)=1/(2√(,2))

tan(θ)=√(,2)/4

5. Find the reciprocal functions csc(θ) sec(θ) and cot(θ) by inverting the values of sin(θ) cos(θ) and tan(θ)

csc(θ)=1/sin(θ)=3

sec(θ)=1/cos(θ)=3/(2√(,2))=(3√(,2))/4

cot(θ)=1/tan(θ)=2√(,2)

Final Answer

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

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