Anonymous
Anonymous asked in Science & MathematicsMathematics · 6 months ago

If cos x +cos y=1, sec x+sec y=4, where 0°<x<180°, find value of x+y.?

Given answer: 120°.

1 Answer

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  • Mewtwo
    Lv 5
    6 months ago
    Best answer

    Using the second equation:

    4 = sec x + sec y

    = 1 / cos x + 1 / cos y

    = (cos y + cos x) / (cos x cos y)

    = 1 / (cos x cos y), or

    cos x cos y = 1/4.

    Using the first equation, cos y = 1 - cos x. Plugging this into the equivalent form of the second equation that we found, then,

    cos x(1 - cos x) = 1/4, or

    4 cos x(1 - cos x) = 1.

    Thus,

    4 cos x - 4 cos^2 x = 1

    4 cos^2 x - 4 cos x + 1 = 0

    Factoring this gives

    (2 cos x - 1)^2 = 0.

    Thus,

    2 cos x - 1 = 0

    cos x = 1/2.

    Since 0° < x < 180°, we verify then that x = 60° (Use your unit circle or a calculator).

    Thus,

    cos y = 1 - cos x

    = 1 - cos 60°

    = 1 - 1/2

    = 1/2.

    So, y = 60° as well. Therefore, x + y = 60° + 60° = 120°.

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