Find the Other Trig Values in Quadrant II cos(x)=-2/3

Problem

cos(x)=−2/3, Quadrant II

Solution

1. Identify the given information and the quadrant. We are given cos(x)=−2/3 in Quadrant II, where sin(x) is positive and tan(x) is negative.

2. Use the Pythagorean identity sin^2(x)+cos^2(x)=1 to find sin(x)

sin^2(x)+(−2/3)^2=1

3. Solve for sin^2(x) by squaring the fraction and subtracting it from 1.

sin^2(x)+4/9=1

sin^2(x)=1−4/9

sin^2(x)=5/9

4. Take the square root and choose the positive value because the angle is in Quadrant II.

sin(x)=√(,5/9)

sin(x)=√(,5)/3

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

tan(x)=√(,5)/3/(−2/3)

tan(x)=−√(,5)/2

6. Find the reciprocal functions sec(x) csc(x) and cot(x) by inverting the primary trigonometric ratios.

sec(x)=1/cos(x)=−3/2

csc(x)=1/sin(x)=3/√(,5)=(3√(,5))/5

cot(x)=1/tan(x)=−2/√(,5)=−(2√(,5))/5

Final Answer

sin(x)=√(,5)/3,tan(x)=−√(,5)/2,csc(x)=(3√(,5))/5,sec(x)=−3/2,cot(x)=−(2√(,5))/5

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