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How do you evaluate #cos^-1(sin(-pi/6))# without a calculator?

Answer 1

#+- (2pi)/3#

Use trig table and unit circle --> #sin (- pi/6) = - 1/2# #cos ^-1 (-1/2) = arccos x = arccos (-1/2)# --> #x = +- (2pi)/3#

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Answer 2

Josiah Anderson

To evaluate ( \cos^{-1}(\sin(-\frac{\pi}{6})) ) without a calculator:

First, note that ( \sin(-\frac{\pi}{6}) = -\sin(\frac{\pi}{6}) = -\frac{1}{2} ).
Recall that the cosine function is positive in the first and fourth quadrants and negative in the second and third quadrants.
Since ( \sin(-\frac{\pi}{6}) = -\frac{1}{2} ), we're dealing with the fourth quadrant where cosine is positive.
In the fourth quadrant, ( \cos^{-1}(x) ) is the angle whose cosine is ( x ) and lies in the range ( [0, \pi] ).
The angle whose cosine is ( -\frac{1}{2} ) in the fourth quadrant is ( \frac{2\pi}{3} ).

So, ( \cos^{-1}(\sin(-\frac{\pi}{6})) = \frac{2\pi}{3} ).

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Answer from HIX Tutor

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.