Find the Other Trig Values in Quadrant III cos(x)=-1/3

Problem

cos(x)=−1/3,π<x<(3*π)/2

Solution

1. Identify the given information and the quadrant. We are given cos(x)=−1/3 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 sin(x)

sin2(x)+(−1/3)2=1

sin2(x)+1/9=1

sin2(x)=8/9

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

sin(x)=−√(,8/9)

sin(x)=−(2√(,2))/3

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

tan(x)=(−(2√(,2))/3)/(−1/3)

tan(x)=2√(,2)

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

sec(x)=1/cos(x)=−3

csc(x)=1/sin(x)=−3/(2√(,2))=−(3√(,2))/4

cot(x)=1/tan(x)=1/(2√(,2))=√(,2)/4

Final Answer

sin(x)=−(2√(,2))/3,tan(x)=2√(,2),csc(x)=−(3√(,2))/4,sec(x)=−3,cot(x)=√(,2)/4

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