mathematics

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evaluate each of the following...sin 75-cos 105-sin 15+cos 15

  • mathematics -

    I assume you want "exact" answers for these, the usual kind of request for these types of questions.

    All your given angles are combinations of the standard angles of 30, 45, 60 etc
    e.g. 75 = 45+30
    15 = 45-30
    105 = 60+45

    You should also have memorized or have quick access to the trig ratios of these
    e.g. sin 45° = √2/2 , cos 60° = 1/2 , etc


    evaluate each one, the do the arithmetic

    e.g.
    sin75 = sin(45+30)
    = sin45cos30 + cos45sin30
    = √2/2(√3/2) + √2/2(1/2)
    = √6/4 + √2/4 = (√6 + √2)/4

    carefully do the others, easy to make errors with those radicals

  • mathematics -

    You could also have done some preliminaray simplification using the complementary angle properties
    e.g.

    cos15° = sin75°
    cos105 = sin(-15) = -sin15

    so sin 75-cos 105-sin 15+cos 15
    sin75 - (-sin15) + sin15 + sin75
    = 2sin75 + 2sin15
    = 2(sin75+sin15)
    Now you only have to do two of the calculations

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