Find the Other Trig Values in Quadrant I sin(x)=24/25

Problem

sin(x)=24/25,0<x<π/2

Solution

1. Identify the given information and the quadrant. Since x is in Quadrant I, all trigonometric ratios (sin() cos() tan() csc() sec() cot() will be positive.

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

(24/25)2+cos2(x)=1

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

576/625+cos2(x)=1

cos2(x)=1−576/625

cos2(x)=49/625

4. Take the square root of both sides. Since x is in Quadrant I, we take the positive root.

cos(x)=√(,49/625)

cos(x)=7/25

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

tan(x)=(24/25)/(7/25)

tan(x)=24/7

6. Find the reciprocal identities for the remaining three functions: csc(x)=1/sin(x) sec(x)=1/cos(x) and cot(x)=1/tan(x)

csc(x)=25/24

sec(x)=25/7

cot(x)=7/24

Final Answer

cos(x)=7/25,tan(x)=24/7,csc(x)=25/24,sec(x)=25/7,cot(x)=7/24

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