Find the Other Trig Values in Quadrant I sin(theta)=1/4

Problem

sin(θ)=1/4,θ∈Quadrant I

Solution

1. Identify the given values and the quadrant. In Quadrant I, all trigonometric functions are positive. Given sin(θ)=y/r=1/4 we have y=1 and r=4

2. Calculate the missing side x using the Pythagorean identity x2+y2=r2

x2+1=4

x2+1=16

x2=15

x=√(,15)

3. Determine the value of cos(θ) using the ratio x/r

cos(θ)=√(,15)/4

4. Determine the value of tan(θ) using the ratio y/x

tan(θ)=1/√(,15)

tan(θ)=√(,15)/15

5. Determine the reciprocal functions csc(θ) sec(θ) and cot(θ)

csc(θ)=1/sin(θ)=4

sec(θ)=1/cos(θ)=4/√(,15)=(4√(,15))/15

cot(θ)=1/tan(θ)=√(,15)

Final Answer

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

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