How do you find the exact value of the following using the unit circle 6 cos (11pi/6) - 2 sin^2 (5pi/4)?

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How do you find the exact value of the following using the unit circle 6 cos (11pi/6) - 2 sin^2 (5pi/4)?

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Answer 1 · 2018-03-17T05:44:37

$6 cos (\frac{11 π}{6}) - 2 {sin}^{2} (\frac{5 π}{4}) = 2$ $color(purple)(6 cos ((11pi)/6) - 2 sin ^2 ((5pi)/4) = 2$

Explanation:

enter image source here

From the above diagram,

$\frac{ˆ 11 π}{6}$ $hat (11pi)/6$ is in IV quadrant where cos is positive.

$cos (\frac{11 π}{6}) = cos ((\frac{11 π}{6}) - 2 π) = cos - (\frac{π}{6}) = cos (\frac{π}{6})$ $cos ((11pi)/6) = cos (((11pi)/6) - 2pi) = cos -(pi/6) = cos (pi/6)$

enter image source here

But $cos (\frac{π}{6}) = cos 60 = x = \frac{1}{2}$ $cos (pi/6) = cos 60 = x = 1/2$ $:. color(red)(cos ((11pi)/6) = 1/2$

$hat (5pi)/4$ is in III Quadrant where sin is negative.

$sin ((5pi)/4) = sin (pi + (pi/4)) = - sin (pi/4) = - sin 45$

But $sin (pi/4) = 1/sqrt2$

$:. color(green)(sin ((5pi)/4) = y = -1/sqrt2$

Returning to the given sum,

$6 cos ((11pi)/6) - 2 sin ^2 ((5pi)/4) = 6 * (1/2) - 2 (-(1/sqrt2))^2$

$=> (6 * (1/2)) -(2* (1/2)) = 3 - 1 = 2$

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How do you find the exact value of the following using the unit circle 6 cos (11pi/6) - 2 sin^2 (5pi/4)?

1 Answer

Explanation:

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