Find the Other Trig Values in Quadrant I sin(theta)=12/13

Problem

sin(θ)=12/13,θ∈Quadrant I

Solution

1. Identify the given values for the right triangle. Since sin(θ)=opposite/hypotenuse we have y=12 and r=13

2. Apply the Pythagorean theorem x^2+y^2=r^2 to find the adjacent side x

x^2+12^2=13^2

x^2+144=169

x^2=25

3. Solve for x Since θ is in Quadrant I, x must be positive.

x=√(,25)

x=5

4. Determine the remaining trigonometric ratios using x=5 y=12 and r=13

cos(θ)=x/r=5/13

tan(θ)=y/x=12/5

csc(θ)=r/y=13/12

sec(θ)=r/x=13/5

cot(θ)=x/y=5/12

Final Answer

cos(θ)=5/13,tan(θ)=12/5,csc(θ)=13/12,sec(θ)=13/5,cot(θ)=5/12

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