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

Problem

sin(x)=24/25,x∈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. Apply the Pythagorean identity sin2(x)+cos2(x)=1 to find the value of cos(x)

(24/25)2+cos2(x)=1

3. Solve for cos2(x) by squaring the fraction and subtracting it from 1.

576/625+cos2(x)=1

cos2(x)=1−576/625

cos2(x)=49/625

4. Determine cos(x) by taking the square root. Since x is in Quadrant II, cos(x) must be negative.

cos(x)=−√(,49/625)

cos(x)=−7/25

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

tan(x)=(24/25)/(−7/25)

tan(x)=−24/7

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

csc(x)=1/sin(x)=25/24

sec(x)=1/cos(x)=−25/7

cot(x)=1/tan(x)=−7/24

Final Answer

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

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