Find the Other Trig Values in Quadrant I tan(theta)=8

Problem

tan(θ)=8,0<θ<π/2

Solution

1. Identify the given information and the quadrant. We are given tan(θ)=8/1 and told that θ is in Quadrant I, where all trigonometric functions are positive.

2. Define the sides of a right triangle using the definition tan(θ)=opposite/adjacent Let the opposite side y=8 and the adjacent side x=1

3. Calculate the hypotenuse r using the Pythagorean theorem r=√(,x2+y2)

r=√(,1+8)

r=√(,1+64)

r=√(,65)

4. Determine the sine value using the ratio sin(θ)=y/r

sin(θ)=8/√(,65)

sin(θ)=(8√(,65))/65

5. Determine the cosine value using the ratio cos(θ)=x/r

cos(θ)=1/√(,65)

cos(θ)=√(,65)/65

6. Determine the cosecant value by taking the reciprocal of sine.

csc(θ)=√(,65)/8

7. Determine the secant value by taking the reciprocal of cosine.

sec(θ)=√(,65)

8. Determine the cotangent value by taking the reciprocal of tangent.

cot(θ)=1/8

Final Answer

sin(θ)=(8√(,65))/65,cos(θ)=√(,65)/65,csc(θ)=√(,65)/8,sec(θ)=√(,65),cot(θ)=1/8

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