Find the Other Trig Values in Quadrant III sin(x)=-4/5

Problem

sin(x)=−4/5,π<x<(3*π)/2

Solution

1. Identify the given information and the quadrant. We are given sin(x)=−4/5 in Quadrant III, where both sin(x) and cos(x) are negative, while tan(x) and cot(x) are positive.

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

(−4/5)2+cos2(x)=1

16/25+cos2(x)=1

cos2(x)=1−16/25

cos2(x)=9/25

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

cos(x)=−√(,9/25)

cos(x)=−3/5

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

tan(x)=(−4/5)/(−3/5)

tan(x)=4/3

5. Determine the reciprocal functions csc(x) sec(x) and cot(x) using their definitions.

csc(x)=1/sin(x)=−5/4

sec(x)=1/cos(x)=−5/3

cot(x)=1/tan(x)=3/4

Final Answer

cos(x)=−3/5,tan(x)=4/3,csc(x)=−5/4,sec(x)=−5/3,cot(x)=3/4

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