# How do you evaluate #cos(sin^-1((sqrt3/2))# without a calculator?

Consider an equilateral triangle with sides of length

Remembering that

#sin(pi/3) = sqrt(3)/2#

Since

#sin^(-1)(sqrt(3)/2) = pi/3#

From the same diagram, remembering

#cos(pi/3) = 1/2#

So:

#cos(sin^(-1)(sqrt(3)/2)) = cos(pi/3) = 1/2#

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Starting from:

Take the square root to find:

So:

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To evaluate ( \cos(\sin^{-1}\left(\frac{\sqrt{3}}{2}\right)) ) without a calculator, you can use the properties of trigonometric functions and basic geometric principles.

First, recall that ( \sin^{-1}(x) ) represents the angle whose sine is ( x ). So, ( \sin^{-1}\left(\frac{\sqrt{3}}{2}\right) ) represents the angle whose sine is ( \frac{\sqrt{3}}{2} ). From the unit circle or trigonometric identities, we know that this angle is ( \frac{\pi}{3} ) radians or ( 60^\circ ).

Now, consider a right triangle where one angle is ( \frac{\pi}{3} ) radians or ( 60^\circ ) and the opposite side to this angle has length ( \sqrt{3} ) while the hypotenuse has length ( 2 ) (since ( \sin(\frac{\pi}{3}) = \frac{\sqrt{3}}{2} )).

By using the Pythagorean theorem, we can find the length of the adjacent side of the triangle: [ \text{Adjacent side} = \sqrt{\text{Hypotenuse}^2 - \text{Opposite side}^2} ] [ = \sqrt{2^2 - (\sqrt{3})^2} = \sqrt{4 - 3} = 1 ]

Now, we can evaluate ( \cos(\sin^{-1}\left(\frac{\sqrt{3}}{2}\right)) ), which is the cosine of the angle ( \frac{\pi}{3} ) or ( 60^\circ ). In our triangle, the adjacent side is 1 and the hypotenuse is 2, so: [ \cos(\sin^{-1}\left(\frac{\sqrt{3}}{2}\right)) = \frac{\text{Adjacent side}}{\text{Hypotenuse}} = \frac{1}{2} ]

Therefore, ( \cos(\sin^{-1}\left(\frac{\sqrt{3}}{2}\right)) = \frac{1}{2} ).

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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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