Find the Other Trig Values in Quadrant I sin(x)=60/61

Problem

sin(x)=60/61,0<x<π/2

Solution

1. Identify the given information and the quadrant. We are given sin(x)=60/61 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)

(60/61)2+cos2(x)=1

3. Solve for cos2(x) by squaring the fraction and subtracting it from 1.

3600/3721+cos2(x)=1

cos2(x)=1−3600/3721

cos2(x)=(3721−3600)/3721

cos2(x)=121/3721

4. Take the square root of both sides. Since x is in Quadrant I, cos(x) must be positive.

cos(x)=√(,121/3721)

cos(x)=11/61

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

tan(x)=(60/61)/(11/61)

tan(x)=60/11

6. 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)=61/60

sec(x)=1/cos(x)=61/11

cot(x)=1/tan(x)=11/60

Final Answer

cos(x)=11/61,tan(x)=60/11,csc(x)=61/60,sec(x)=61/11,cot(x)=11/60

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