Question 870531
Since 90 < alpha < 180 and we're told that cos(alpha) = -4/5, we know sin(alpha) = 3/5. Use a right triangle to see this (it would be a 3,4,5 right triangle). Keep in mind that sin(alpha) is positive because we're in quadrant 2.


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If 90 < alpha < 180, then doubling each piece gives us 180 < 2alpha < 360


So 2alpha is in quadrants 3 and 4. 


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a) Find sin(2alpha)



sin(2alpha) = 2*sin(alpha)*cos(alpha)
sin(2alpha) = 2*(3/5)*(-4/5)
sin(2alpha) = -24/25



Because 180 < 2alpha < 360 and sin(2alpha) = -24/25, this means we're still in quadrants 3 and 4 (where sine is negative). Unfortunately we don't have enough info to determine which quadrant 2alpha is in.


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b) Find cos(2alpha)



cos(2alpha) = cos^2(alpha) - sin^2(alpha)
cos(2alpha) = (-4/5)^2 - (3/5)^2
cos(2alpha) = 16/25 - 9/25
cos(2alpha) = 7/25



Because cos(2alpha) is positive and 180 < 2alpha < 360, we now have enough info to conclude that 2alpha is in quadrant 4. This is where cosine is positive. This is assuming we use both pieces of info that sin(2alpha) = -24/25 and cos(2alpha) = 7/25 along with the restriction that 180 < 2alpha < 360