Find the Other Trig Values in Quadrant I sin(x)=1/3

Problem

sin(x)=1/3,x∈Quadrant I

Solution

1. Identify the given information and the quadrant. In Quadrant I, all trigonometric functions (sin(x) cos(x) tan(x) csc(x) sec(x) cot(x) are positive.

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

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

1/9+cos2(x)=1

cos2(x)=1−1/9

cos2(x)=8/9

3. Solve for cos(x) by taking the square root. Since x is in Quadrant I, we take the positive root.

cos(x)=√(,8/9)

cos(x)=(2√(,2))/3

4. Find tan(x) using the ratio tan(x)=sin(x)/cos(x)

tan(x)=(1/3)/(2√(,2)/3)

tan(x)=1/(2√(,2))

tan(x)=√(,2)/4

5. Find the reciprocal functions csc(x) sec(x) and cot(x) by inverting the values of sin(x) cos(x) and tan(x)

csc(x)=1/sin(x)=3

sec(x)=1/cos(x)=3/(2√(,2))=(3√(,2))/4

cot(x)=1/tan(x)=2√(,2)

Final Answer

cos(x)=(2√(,2))/3,tan(x)=√(,2)/4,csc(x)=3,sec(x)=(3√(,2))/4,cot(x)=2√(,2)

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