Find the Other Trig Values in Quadrant I cos(a)=( square root of 11)/11

Problem

cos(a)=√(,11)/11, Quadrant I

Solution

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

2. Apply the Pythagorean identity sin^2(a)+cos^2(a)=1 to find sin(a)

sin^2(a)+(√(,11)/11)^2=1

3. Simplify the squared term and solve for sin^2(a)

sin^2(a)+11/121=1

sin^2(a)+1/11=1

sin^2(a)=1−1/11

sin^2(a)=10/11

4. Take the square root of both sides. Since the angle is in Quadrant I, we take the positive root.

sin(a)=√(,10/11)

sin(a)=√(,110)/11

5. Find the tangent using the identity tan(a)=sin(a)/cos(a)

tan(a)=√(,110)/11/√(,11)/11

tan(a)=√(,110)/√(,11)

tan(a)=√(,10)

6. Find the reciprocal functions csc(a) sec(a) and cot(a)

csc(a)=1/sin(a)=11/√(,110)=(11√(,110))/110=√(,110)/10

sec(a)=1/cos(a)=11/√(,11)=√(,11)

cot(a)=1/tan(a)=1/√(,10)=√(,10)/10

Final Answer

sin(a)=√(,110)/11,tan(a)=√(,10),csc(a)=√(,110)/10,sec(a)=√(,11),cot(a)=√(,10)/10

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