Find the Other Trig Values in Quadrant I sec(theta)=5

Problem

sec(θ)=5,0<θ<π/2

Solution

1. Identify the given trigonometric ratio and the quadrant. Since sec(θ)=5 and θ is in Quadrant I, all trigonometric functions will be positive.

2. Find the cosine by using the reciprocal identity cos(θ)=1/sec(θ)

cos(θ)=1/5

3. Find the sine using the Pythagorean identity sin2(θ)+cos2(θ)=1

sin2(θ)+(1/5)2=1

sin2(θ)+1/25=1

sin2(θ)=24/25

sin(θ)=√(,24/25)

sin(θ)=(2√(,6))/5

4. Find the cosecant by using the reciprocal identity csc(θ)=1/sin(θ)

csc(θ)=5/(2√(,6))

csc(θ)=(5√(,6))/12

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

tan(θ)=(2√(,6))/5/1/5

tan(θ)=2√(,6)

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

cot(θ)=1/(2√(,6))

cot(θ)=√(,6)/12

Final Answer

sin(θ)=(2√(,6))/5,cos(θ)=1/5,tan(θ)=2√(,6),csc(θ)=(5√(,6))/12,cot(θ)=√(,6)/12

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