Find the Other Trig Values in Quadrant I csc(theta)=4

Problem

csc(θ)=4,0<θ<π/2

Solution

1. Identify the given trigonometric ratio and the quadrant. Since csc(θ)=4 and csc(θ)=1/sin(θ) we find the sine value.

sin(θ)=1/4

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

(1/4)2+cos2(θ)=1

3. Solve for cos(θ) Since θ is in Quadrant I, the cosine value must be positive.

1/16+cos2(θ)=1

cos2(θ)=15/16

cos(θ)=√(,15)/4

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

tan(θ)=(1/4)/(√(,15)/4)

tan(θ)=1/√(,15)

tan(θ)=√(,15)/15

5. Determine the reciprocal identities for the remaining functions: sec(θ)=1/cos(θ) and cot(θ)=1/tan(θ)

sec(θ)=4/√(,15)

sec(θ)=(4√(,15))/15

cot(θ)=√(,15)

Final Answer

sin(θ)=1/4,cos(θ)=√(,15)/4,tan(θ)=√(,15)/15,sec(θ)=(4√(,15))/15,cot(θ)=√(,15)

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