# How do you find the exact values of cos(11pi/12) using the half angle formula?

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To find the exact value of (\cos\left(\frac{11\pi}{12}\right)) using the half-angle formula, we can first express (\frac{11\pi}{12}) as a sum or difference of two angles whose cosine values are known. Since (\frac{11\pi}{12} = \frac{22\pi}{24} - \frac{\pi}{24}), we can use the half-angle formula for cosine, which states that (\cos\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1+\cos(\theta)}{2}}).

Now, we find the cosine of (\frac{\pi}{12}) and (\frac{\pi}{24}) using known values or angles that can be expressed in terms of these angles. Once we have these values, we substitute them into the half-angle formula to find the cosine of (\frac{11\pi}{12}).

Let's denote (x = \frac{\pi}{12}). Then, (\cos(2x) = \cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2}). By using the double-angle formula for cosine, we have (\cos(2x) = 2\cos^2(x) - 1). Solving for (\cos(x)), we get (\cos(x) = \sqrt{\frac{1 + \cos(2x)}{2}} = \sqrt{\frac{1 + \frac{\sqrt{3}}{2}}{2}}).

Now, we also need (\cos\left(\frac{\pi}{24}\right)). Let (y = \frac{\pi}{24}). Then, (\cos(2y) = \cos\left(\frac{\pi}{12}\right)). Using the double-angle formula again, we have (\cos(2y) = 2\cos^2(y) - 1). Solving for (\cos(y)), we find (\cos(y) = \sqrt{\frac{1 + \cos(2y)}{2}} = \sqrt{\frac{1 + \cos\left(\frac{\pi}{12}\right)}{2}}).

Finally, using the angle addition formula for cosine, (\cos\left(\frac{11\pi}{12}\right) = \cos\left(\frac{\pi}{12} + \frac{\pi}{24}\right) = \cos(x + y) = \cos(x)\cos(y) - \sin(x)\sin(y)). Substitute the values of (\cos(x)), (\cos(y)), (\sin(x)), and (\sin(y)), and calculate to find the exact value of (\cos\left(\frac{11\pi}{12}\right)).

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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