# How do you evaluate the expression #cos(u+v)# given #cosu=4/7# with #0<u<pi/2# and #sinv=-9/10# with #pi<v<(3pi)/2#?

Let's compute (1) cos v and (2) sin u:

Then

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Given ( \cos(u) = \frac{4}{7} ) with ( 0 < u < \frac{\pi}{2} ) and ( \sin(v) = -\frac{9}{10} ) with ( \pi < v < \frac{3\pi}{2} ), we need to evaluate ( \cos(u + v) ).

We'll use the trigonometric identities:

- ( \cos(u + v) = \cos(u)\cos(v) - \sin(u)\sin(v) )

First, we need to find ( \sin(u) ). Since ( \cos(u) = \frac{4}{7} ) and ( 0 < u < \frac{\pi}{2} ), we can use the Pythagorean identity:

[ \sin^2(u) + \cos^2(u) = 1 ] [ \sin^2(u) + \left(\frac{4}{7}\right)^2 = 1 ] [ \sin^2(u) + \frac{16}{49} = 1 ] [ \sin^2(u) = 1 - \frac{16}{49} = \frac{33}{49} ] [ \sin(u) = \pm \sqrt{\frac{33}{49}} = \pm \frac{\sqrt{33}}{7} ]

Since ( 0 < u < \frac{\pi}{2} ), ( \sin(u) ) must be positive. So, ( \sin(u) = \frac{\sqrt{33}}{7} ).

Now, we have ( \cos(u) ), ( \sin(u) ), and ( \sin(v) ). We need to find ( \cos(v) ).

Since ( \sin(v) = -\frac{9}{10} ) and ( \pi < v < \frac{3\pi}{2} ), we can use the fact that ( \cos(v) ) is negative in the third quadrant:

[ \cos^2(v) + \sin^2(v) = 1 ] [ \cos^2(v) + \left(-\frac{9}{10}\right)^2 = 1 ] [ \cos^2(v) + \frac{81}{100} = 1 ] [ \cos^2(v) = 1 - \frac{81}{100} = \frac{19}{100} ] [ \cos(v) = -\sqrt{\frac{19}{100}} = -\frac{\sqrt{19}}{10} ]

Now, substitute the values into the formula for ( \cos(u + v) ):

[ \cos(u + v) = \frac{4}{7} \left( -\frac{\sqrt{19}}{10} \right) - \frac{\sqrt{33}}{7} \left( -\frac{9}{10} \right) ]

[ \cos(u + v) = -\frac{4\sqrt{19}}{70} + \frac{9\sqrt{33}}{70} ]

[ \cos(u + v) = \frac{-4\sqrt{19} + 9\sqrt{33}}{70} ]

So, ( \cos(u + v) = \frac{-4\sqrt{19} + 9\sqrt{33}}{70} ).

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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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