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How do you evaluate #tan(arccos(-1/3))#?

Answer 1

Daniel Acosta

# -sqrt 10#

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Answer 2

Jordan Armstrong

To evaluate ( \tan(\arccos(-1/3)) ), we first find the cosine inverse of ( -1/3 ), which is an angle whose cosine is ( -1/3 ). Since cosine is the ratio of the adjacent side to the hypotenuse in a right triangle, we construct a triangle where the adjacent side is ( -1 ) and the hypotenuse is ( 3 ). The opposite side can be found using the Pythagorean theorem, which gives ( \sqrt{3^2 - (-1)^2} = \sqrt{9 - 1} = \sqrt{8} = 2\sqrt{2} ). Therefore, the angle whose cosine is ( -1/3 ) is ( \arccos(-1/3) \approx 109.47^\circ ).

Now, we can use the tangent function, which is the ratio of the opposite side to the adjacent side in a right triangle, to find ( \tan(\arccos(-1/3)) ). So, ( \tan(\arccos(-1/3)) = \frac{2\sqrt{2}}{-1} = -2\sqrt{2} ).

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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.