math

If Ѳ is in Quadrant II, and cos Ѳ=-3/4 , find an exact value for sin 2Ѳ.

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  1. Make a sketch of your triangle in quadrant II to see that
    x = -3, r = 4 and from Pythagoras that y = √7 in II

    so sinØ = √7/4

    sin 2Ø = 2sinØcosØ
    = 2((√7/4)(-3/4)
    = -3√7/8

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  2. sin2A= 2sinAcosA

    now we know that cosA=-3/4, and theta is in the second quadrant so you sketch a triangle in the second quadrant, remember the bowtie method

    -3 is a leg and 4 is the hypotenuse based on sohcahtoa

    plug these two into the pythagorean theorem to find the missing leg.

    then you know that the missing leg is sqrt of 7

    so sinA is sqrt7/4

    2sinAcosA= sin2A

    2(sqrt7/4)(-3/4)= sin2A

    = -6(sqrt7)/16
    = -3(sqrt7)/8

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