Find the Other Trig Values in Quadrant II cos(x)=-24/25

Problem

cos(x)=−24/25, Quadrant II

Solution

1. Identify the given information and the signs of trigonometric functions in Quadrant II. In Quadrant II, sin(x) and csc(x) are positive, while cos(x) sec(x) tan(x) and cot(x) are negative.

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

sin^2(x)+(−24/25)^2=1

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

sin^2(x)+576/625=1

sin^2(x)=1−576/625

sin^2(x)=49/625

4. Take the square root and choose the positive value because sin(x) is positive in Quadrant II.

sin(x)=√(,49/625)

sin(x)=7/25

5. Find the reciprocal functions sec(x) and csc(x) using the definitions sec(x)=1/cos(x) and csc(x)=1/sin(x)

sec(x)=−25/24

csc(x)=25/7

6. Find the quotient functions tan(x) and cot(x) using the definitions tan(x)=sin(x)/cos(x) and cot(x)=1/tan(x)

tan(x)=(7/25)/(−24/25)

tan(x)=−7/24

cot(x)=−24/7

Final Answer

sin(x)=7/25,tan(x)=−7/24,csc(x)=25/7,sec(x)=−25/24,cot(x)=−24/7

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