# A triangle has sides A, B, and C. The angle between sides A and B is #(2pi)/3#. If side C has a length of #2 # and the angle between sides B and C is #pi/12#, what is the length of side A?

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To find the length of side A in the triangle, you can use the Law of Cosines. The formula is:

[ C^2 = A^2 + B^2 - 2AB \cdot \cos(C) ]

Given that side C has a length of 2, and the angle between sides B and C is ( \frac{\pi}{12} ), we have:

[ 2^2 = A^2 + B^2 - 2AB \cdot \cos\left(\frac{\pi}{12}\right) ]

We also know that the angle between sides A and B is ( \frac{2\pi}{3} ), which is ( 120^\circ ) in degrees.

Now, we can solve for side A using the Law of Cosines equation. After rearranging, we get:

[ A^2 = 2^2 + B^2 - 2 \cdot 2B \cdot \cos\left(\frac{\pi}{12}\right) ]

[ A^2 = 4 + B^2 - 4B \cdot \cos\left(\frac{\pi}{12}\right) ]

Since the angle between sides A and B is ( \frac{2\pi}{3} ), which is ( 120^\circ ), we know ( \cos\left(\frac{2\pi}{3}\right) = -\frac{1}{2} ).

Thus, we can replace ( \cos\left(\frac{\pi}{12}\right) ) with ( -\frac{1}{2} ) and solve for side A:

[ A^2 = 4 + B^2 - 4B \cdot \left(-\frac{1}{2}\right) ]

[ A^2 = 4 + B^2 + 2B ]

Given side C has a length of 2, using Law of Sines, we can find B:

[ \frac{\sin(C)}{C} = \frac{\sin(B)}{B} ]

[ \frac{\sin\left(\frac{\pi}{12}\right)}{2} = \frac{\sin(B)}{B} ]

[ \sin(B) = \frac{B \cdot \sin\left(\frac{\pi}{12}\right)}{2} ]

[ B = \frac{2 \cdot \sin\left(\frac{\pi}{12}\right)}{\sin(B)} ]

[ B \approx \frac{2 \cdot 0.2588}{\sin(B)} ]

[ B \approx \frac{0.5176}{\sin(B)} ]

[ B \approx \frac{0.5176}{\sin(B)} ]

[ B \approx \frac{0.5176}{0.2588} ]

[ B \approx 2 ]

Now substitute B = 2 into the equation for A:

[ A^2 = 4 + 2^2 + 2(2) ]

[ A^2 = 4 + 4 + 4 ]

[ A^2 = 12 ]

[ A = \sqrt{12} ]

[ A \approx 3.4641 ]

So, the length of side A is approximately 3.4641.

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You can use the Law of Cosines to find the length of side A. The formula is:

[ A^2 = B^2 + C^2 - 2BC \cdot \cos(\theta) ]

Where:

- ( A ) is the length of side A
- ( B ) is the length of side B
- ( C ) is the length of side C
- ( \theta ) is the angle between sides B and C

Substitute the given values into the formula and solve for ( A ).

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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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