Find the Other Trig Values in Quadrant III sin(x)=-15/17

Problem

sin(x)=−15/17,π<x<(3*π)/2

Solution

1. Identify the given information and the quadrant. We are given sin(x)=−15/17 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 cos(x)

(−15/17)2+cos2(x)=1

225/289+cos2(x)=1

cos2(x)=1−225/289

cos2(x)=64/289

3. Solve for cos(x) by taking the square root. Since x is in Quadrant III, cos(x) must be negative.

cos(x)=−√(,64/289)

cos(x)=−8/17

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

tan(x)=(−15/17)/(−8/17)

tan(x)=15/8

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

csc(x)=−17/15

sec(x)=−17/8

cot(x)=8/15

Final Answer

cos(x)=−8/17,tan(x)=15/8,csc(x)=−17/15,sec(x)=−17/8,cot(x)=8/15

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