Find the Other Trig Values in Quadrant I sin(x)=9/41

Problem

sin(x)=9/41, Quadrant I

Solution

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

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

(9/41)2+cos2(x)=1

81/1681+cos2(x)=1

cos2(x)=1−81/1681

cos2(x)=1600/1681

cos(x)=√(,1600/1681)

cos(x)=40/41

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

tan(x)=(9/41)/(40/41)

tan(x)=9/40

4. 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)=41/9

sec(x)=1/cos(x)=41/40

cot(x)=1/tan(x)=40/9

Final Answer

cos(x)=40/41,tan(x)=9/40,csc(x)=41/9,sec(x)=41/40,cot(x)=40/9

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