Find the Other Trig Values in Quadrant I csc(x)=5/3

Problem

csc(x)=5/3,x∈Quadrant I

Solution

1. Identify the relationship between the cosecant function and the sine function.

sin(x)=1/csc(x)

2. Substitute the given value to find the sine of x

sin(x)=3/5

3. Apply the Pythagorean identity to find the cosine of x

sin2(x)+cos2(x)=1

4. Substitute the sine value into the identity.

(3/5)2+cos2(x)=1

5. Solve for cos2(x)

9/25+cos2(x)=1

cos2(x)=1−9/25

cos2(x)=16/25

6. Take the square root and choose the positive value because x is in Quadrant I.

cos(x)=√(,16/25)

cos(x)=4/5

7. Calculate the tangent of x using the ratio of sine to cosine.

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

tan(x)=(3/5)/(4/5)

tan(x)=3/4

8. Calculate the cotangent of x as the reciprocal of tangent.

cot(x)=1/tan(x)

cot(x)=4/3

9. Calculate the secant of x as the reciprocal of cosine.

sec(x)=1/cos(x)

sec(x)=5/4

Final Answer

sin(x)=3/5,cos(x)=4/5,tan(x)=3/4,sec(x)=5/4,cot(x)=4/3

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