Find the Other Trig Values in Quadrant IV sin(x)=-( square root of 3)/2

Problem

sin(x)=−√(,3)/2, Quadrant IV

Solution

1. Identify the given information and the quadrant. In Quadrant IV, x coordinates are positive and y coordinates are negative, meaning cos(x)>0 and sin(x)<0

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

(−√(,3)/2)2+cos2(x)=1

3/4+cos2(x)=1

cos2(x)=1−3/4

cos2(x)=1/4

3. Solve for cos(x) by taking the square root. Since x is in Quadrant IV, cos(x) must be positive.

cos(x)=√(,1/4)

cos(x)=1/2

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

tan(x)=(−√(,3)/2)/1/2

tan(x)=−√(,3)

5. Determine the reciprocal trigonometric functions csc(x) sec(x) and cot(x)

csc(x)=1/sin(x)=−2/√(,3)=−(2√(,3))/3

sec(x)=1/cos(x)=2

cot(x)=1/tan(x)=−1/√(,3)=−√(,3)/3

Final Answer

cos(x)=1/2,tan(x)=−√(,3),csc(x)=−(2√(,3))/3,sec(x)=2,cot(x)=−√(,3)/3

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