Find the Other Trig Values in Quadrant I tan(theta)=0.74

Problem

tan(θ)=0.74,θ∈Quadrant I

Solution

1. Convert the decimal value of the tangent function to a fraction to identify the opposite and adjacent sides of a right triangle.

tan(θ)=74/100=37/50

2. Identify the lengths of the legs of the right triangle where the opposite side y=37 and the adjacent side x=50

y=37

x=50

3. Apply the Pythagorean theorem to find the hypotenuse r

r=√(,x2+y2)

r=√(,50+37)

r=√(,2500+1369)

r=√(,3869)

4. Determine the sine and cosine values using the ratios sin(θ)=y/r and cos(θ)=x/r Since θ is in Quadrant I, all values are positive.

sin(θ)=37/√(,3869)

cos(θ)=50/√(,3869)

5. Determine the reciprocal trigonometric values by inverting the sine, cosine, and tangent ratios.

csc(θ)=√(,3869)/37

sec(θ)=√(,3869)/50

cot(θ)=50/37

Final Answer

sin(θ)=(37√(,3869))/3869,cos(θ)=(50√(,3869))/3869,csc(θ)=√(,3869)/37,sec(θ)=√(,3869)/50,cot(θ)=50/37

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