Find the Other Trig Values in Quadrant I tan(theta)=4/3

Problem

tan(θ)=4/3,θ∈Quadrant I

Solution

1. Identify the components of the tangent function. In a right triangle, tan(θ)=opposite/adjacent Given tan(θ)=4/3 let the opposite side y=4 and the adjacent side x=3

2. Calculate the hypotenuse r using the Pythagorean theorem x2+y2=r2

3+4=r2

9+16=r2

25=r2

r=5

3. Determine the sine and cosine values. Since θ is in Quadrant I, all trigonometric ratios are positive.

sin(θ)=y/r=4/5

cos(θ)=x/r=3/5

4. Find the reciprocal trigonometric values by inverting the primary ratios.

csc(θ)=1/sin(θ)=5/4

sec(θ)=1/cos(θ)=5/3

cot(θ)=1/tan(θ)=3/4

Final Answer

sin(θ)=4/5,cos(θ)=3/5,csc(θ)=5/4,sec(θ)=5/3,cot(θ)=3/4

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