Find the Other Trig Values in Quadrant I tan(theta)=4

Problem

tan(θ)=4,θ∈Quadrant I

Solution

1. Identify the given information and the quadrant. Since θ is in Quadrant I, all trigonometric functions (sin() cos() tan() csc() sec() cot() will be positive.

2. Use the definition of the tangent function in a right triangle, where tan(θ)=opposite/adjacent We can let the opposite side y=4 and the adjacent side x=1

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

r=√(,1+4)

r=√(,1+16)

r=√(,17)

4. Determine the sine and cosine values using the ratios sin(θ)=y/r and cos(θ)=x/r

sin(θ)=4/√(,17)=(4√(,17))/17

cos(θ)=1/√(,17)=√(,17)/17

5. Find the reciprocal trigonometric values.

cot(θ)=1/tan(θ)=1/4

sec(θ)=1/cos(θ)=√(,17)

csc(θ)=1/sin(θ)=√(,17)/4

Final Answer

sin(θ)=(4√(,17))/17,cos(θ)=√(,17)/17,cot(θ)=1/4,sec(θ)=√(,17),csc(θ)=√(,17)/4

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