Find the Other Trig Values in Quadrant I sec(theta)=7

Problem

sec(θ)=7,0<θ<π/2

Solution

1. Identify the given trigonometric ratio and the quadrant. Since sec(θ)=7 and θ is in Quadrant I, all trigonometric values will be positive.

2. Determine the cosine value using the reciprocal identity cos(θ)=1/sec(θ)

cos(θ)=1/7

3. Apply the Pythagorean identity sin2(θ)+cos2(θ)=1 to find sin(θ)

sin2(θ)+(1/7)2=1

sin2(θ)+1/49=1

sin2(θ)=48/49

sin(θ)=√(,48/49)

sin(θ)=(4√(,3))/7

4. Calculate the tangent value using the identity tan(θ)=sin(θ)/cos(θ)

tan(θ)=(4√(,3))/7/1/7

tan(θ)=4√(,3)

5. Find the cosecant value using the reciprocal identity csc(θ)=1/sin(θ)

csc(θ)=7/(4√(,3))

csc(θ)=(7√(,3))/12

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

cot(θ)=1/(4√(,3))

cot(θ)=√(,3)/12

Final Answer

sin(θ)=(4√(,3))/7,cos(θ)=1/7,tan(θ)=4√(,3),csc(θ)=(7√(,3))/12,cot(θ)=√(,3)/12

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