Finding the value of sin θ given cos θ = 12/13 using the Pythagorean Theorem

To find sin θ given that cos θ = 12/13, we can use the Pythagorean identity:

[ sin^2 θ + cos^2 θ = 1 ]

First, we calculate cos² θ:

[ cos^2 θ = left( frac{12}{13} right)^2 = frac{144}{169} ]

Now, substitute this value into the Pythagorean identity:

[ sin^2 θ + frac{144}{169} = 1 ]

To isolate sin² θ, subtract cos² θ from both sides:

[ sin^2 θ = 1 – frac{144}{169} ]

Convert 1 to a fraction with a denominator of 169:

[ 1 = frac{169}{169} ]

Now subtract:

[ sin^2 θ = frac{169}{169} – frac{144}{169} = frac{25}{169} ]

Now, take the square root of both sides to find sin θ:

[ sin θ = sqrt{frac{25}{169}} = frac{5}{13} ]

Thus, the correct answer is:

A. sin θ = 5/13