Find the Other Trig Values in Quadrant I sec(x)=4/3

Problem

sec(x)=4/3, Quadrant I

Solution

1. Identify the relationship between the secant function and the cosine function.

cos(x)=1/sec(x)

2. Substitute the given value to find cos(x)

cos(x)=3/4

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

sin2(x)+cos2(x)=1

4. Substitute the value of cos(x) into the identity.

sin2(x)+(3/4)2=1

5. Solve for sin(x) noting that in Quadrant I, all trigonometric values are positive.

sin2(x)=1−9/16

sin2(x)=7/16

sin(x)=√(,7)/4

6. Find csc(x) by taking the reciprocal of sin(x)

csc(x)=1/sin(x)

csc(x)=4/√(,7)

csc(x)=(4√(,7))/7

7. Find tan(x) using the ratio of sine to cosine.

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

tan(x)=√(,7)/4/3/4

tan(x)=√(,7)/3

8. Find cot(x) by taking the reciprocal of tan(x)

cot(x)=1/tan(x)

cot(x)=3/√(,7)

cot(x)=(3√(,7))/7

Final Answer

sin(x)=√(,7)/4,cos(x)=3/4,tan(x)=√(,7)/3,csc(x)=(4√(,7))/7,cot(x)=(3√(,7))/7

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