Find the Other Trig Values in Quadrant I csc(x)=3/2

Problem

csc(x)=3/2,x∈Quadrant I

Solution

1. Identify the relationship between the cosecant function and the sine function using the reciprocal identity csc(x)=1/sin(x)

sin(x)=2/3

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

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

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

4/9+cos2(x)=1

cos2(x)=1−4/9

cos2(x)=5/9

4. Determine the value of cos(x) by taking the square root. Since x is in Quadrant I, the cosine value must be positive.

cos(x)=√(,5/9)

cos(x)=√(,5)/3

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

tan(x)=(2/3)/(√(,5)/3)

tan(x)=2/√(,5)

tan(x)=(2√(,5))/5

6. Find the cotangent value using the reciprocal identity cot(x)=1/tan(x)

cot(x)=√(,5)/2

7. Find the secant value using the reciprocal identity sec(x)=1/cos(x)

sec(x)=3/√(,5)

sec(x)=(3√(,5))/5

Final Answer

sin(x)=2/3,cos(x)=√(,5)/3,tan(x)=(2√(,5))/5,sec(x)=(3√(,5))/5,cot(x)=√(,5)/2

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