Find the Other Trig Values in Quadrant I sin(x)=1/5

Problem

sin(x)=1/5,0<x<π/2

Solution

1. Identify the given information and the quadrant. We are given sin(x)=1/5 in Quadrant I, where all trigonometric functions are positive.

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

(1/5)2+cos2(x)=1

3. Solve for cos(x) by subtracting the squared sine value from 1 and taking the positive square root.

cos2(x)=1−1/25

cos2(x)=24/25

cos(x)=√(,24/25)

cos(x)=(2√(,6))/5

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

tan(x)=(1/5)/(2√(,6)/5)

tan(x)=1/(2√(,6))

tan(x)=√(,6)/12

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

csc(x)=1/sin(x)=5

sec(x)=1/cos(x)=5/(2√(,6))=(5√(,6))/12

cot(x)=1/tan(x)=2√(,6)

Final Answer

cos(x)=(2√(,6))/5,tan(x)=√(,6)/12,csc(x)=5,sec(x)=(5√(,6))/12,cot(x)=2√(,6)

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