# What is the slope of the line normal to the tangent line of #f(x) = cosx+sin(2x-pi/12) # at # x= (5pi)/8 #?

Slope

Slope

For the slope of the normal line

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To find the slope of the line normal to the tangent line of the function ( f(x) = \cos(x) + \sin(2x - \frac{\pi}{12}) ) at ( x = \frac{5\pi}{8} ), we first find the derivative of the function and evaluate it at ( x = \frac{5\pi}{8} ). Then, we find the negative reciprocal of this derivative to get the slope of the normal line.

The derivative of ( f(x) ) is ( f'(x) = -\sin(x) + 2\cos(2x - \frac{\pi}{12}) ).

Evaluate ( f'(x) ) at ( x = \frac{5\pi}{8} ): [ f'\left(\frac{5\pi}{8}\right) = -\sin\left(\frac{5\pi}{8}\right) + 2\cos\left(2\left(\frac{5\pi}{8}\right) - \frac{\pi}{12}\right) ] [ f'\left(\frac{5\pi}{8}\right) = -\sin\left(\frac{5\pi}{8}\right) + 2\cos\left(\frac{5\pi}{4} - \frac{\pi}{12}\right) ] [ f'\left(\frac{5\pi}{8}\right) = -\sin\left(\frac{5\pi}{8}\right) + 2\cos\left(\frac{15\pi}{12} - \frac{\pi}{12}\right) ] [ f'\left(\frac{5\pi}{8}\right) = -\sin\left(\frac{5\pi}{8}\right) + 2\cos\left(\frac{14\pi}{12}\right) ] [ f'\left(\frac{5\pi}{8}\right) = -\sin\left(\frac{5\pi}{8}\right) + 2\cos\left(\frac{7\pi}{6}\right) ]

Using the unit circle, ( \sin\left(\frac{5\pi}{8}\right) = \sin\left(\frac{\pi}{8}\right) ) and ( \cos\left(\frac{7\pi}{6}\right) = -\frac{\sqrt{3}}{2} ).

[ f'\left(\frac{5\pi}{8}\right) = -\sin\left(\frac{\pi}{8}\right) + 2\left(-\frac{\sqrt{3}}{2}\right) ] [ f'\left(\frac{5\pi}{8}\right) = -\sin\left(\frac{\pi}{8}\right) - \sqrt{3} ]

The slope of the tangent line at ( x = \frac{5\pi}{8} ) is ( f'\left(\frac{5\pi}{8}\right) = -\sin\left(\frac{\pi}{8}\right) - \sqrt{3} ).

The slope of the normal line is the negative reciprocal of the slope of the tangent line: [ \text{Slope of normal line} = -\frac{1}{f'\left(\frac{5\pi}{8}\right)} = -\frac{1}{-\sin\left(\frac{\pi}{8}\right) - \sqrt{3}} ]

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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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