Find the Other Trig Values in Quadrant II cos(x)=-2/7

Problem

cos(x)=−2/7, Quadrant II

Solution

1. Identify the given information and the quadrant. We are given cos(x)=−2/7 in Quadrant II, where sin(x) is positive and tan(x) is negative.

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

sin2(x)+(−2/7)2=1

3. Solve for sin2(x) by squaring the fraction and subtracting it from 1.

sin2(x)+4/49=1

sin2(x)=45/49

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

sin(x)=√(,45/49)

sin(x)=(3√(,5))/7

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

tan(x)=(3√(,5))/7/(−2/7)

tan(x)=−(3√(,5))/2

6. Find the reciprocal functions sec(x) csc(x) and cot(x) by inverting the values of cos(x) sin(x) and tan(x)

sec(x)=1/cos(x)=−7/2

csc(x)=1/sin(x)=7/(3√(,5))=(7√(,5))/15

cot(x)=1/tan(x)=−2/(3√(,5))=−(2√(,5))/15

Final Answer

sin(x)=(3√(,5))/7,tan(x)=−(3√(,5))/2,sec(x)=−7/2,csc(x)=(7√(,5))/15,cot(x)=−(2√(,5))/15

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