Given cos θ = 12/13, use the Pythagorean Theorem to find the value of sin θ.
A. sin θ = 5/13
B. sin θ = 5/12
C. sin θ = 13/12
D. sin θ = 12/13
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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