SOLUTION: Please help me solve this two part question: Given cos alpha= -4/5, 90 < alpha < 180 Then find a) sin 2 alpha b) cos 2 alpha

Question 870531: Please help me solve this two part question: Given cos alpha= -4/5, 90 < alpha < 180
Then find a) sin 2 alpha
b) cos 2 alpha
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
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
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