Find the Other Trig Values in Quadrant IV sec(x)=3

Problem

sec(x)=3, Quadrant IV

Solution

1. Identify the given information and the quadrant. We are given sec(x)=3 which means cos(x)=1/3 In Quadrant IV, xcoordinates are positive and ycoordinates are negative.

2. Determine the cosine value using the reciprocal identity.

cos(x)=1/sec(x)

cos(x)=1/3

3. Apply the Pythagorean identity to find sin(x)

sin2(x)+cos2(x)=1

sin2(x)+(1/3)2=1

sin2(x)+1/9=1

sin2(x)=8/9

4. Solve for sin(x) by taking the square root. Since x is in Quadrant IV, the sine value must be negative.

sin(x)=−√(,8/9)

sin(x)=−(2√(,2))/3

5. Calculate csc(x) using the reciprocal identity.

csc(x)=1/sin(x)

csc(x)=−3/(2√(,2))

csc(x)=−(3√(,2))/4

6. Calculate tan(x) using the quotient identity.

tan(x)=sin(x)/cos(x)

tan(x)=(−2√(,2)/3)/(1/3)

tan(x)=−2√(,2)

7. Calculate cot(x) using the reciprocal identity.

cot(x)=1/tan(x)

cot(x)=−1/(2√(,2))

cot(x)=−√(,2)/4

Final Answer

sin(x)=−(2√(,2))/3,cos(x)=1/3,tan(x)=−2√(,2),csc(x)=−(3√(,2))/4,cot(x)=−√(,2)/4

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