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

Problem

sin(x)=−24/25,π<x<(3*π)/2

Solution

1. Identify the given information and the quadrant. We are given sin(x)=−24/25 in Quadrant III, where both sin(x) and cos(x) are negative, while tan(x) and cot(x) are 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 cos2(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 and choose the negative sign because the angle x is in Quadrant III.

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 values for the remaining trigonometric 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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