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

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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1 Answer

  1. 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

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