# How do you solve #arcsin(sqrt(2x))=arccos(sqrtx)#?

We have to take the sine or the cosine of both sides. Pro Tip: choose cosine. It probably doesn't matter here, but it's a good rule.

Now let's do the problem

Check:

Let's take sines this time.

Clearly the positive principal value of the arccos leads to a positive sine.

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To solve the equation arcsin(sqrt(2x)) = arccos(sqrt(x)), first, express both sides in terms of the same trigonometric function, then apply trigonometric identities and solve for x.

arcsin(sqrt(2x)) = arccos(sqrt(x)) sin(arcsin(sqrt(2x))) = sin(arccos(sqrt(x))) sqrt(2x) = cos(arccos(sqrt(x))) sqrt(2x) = sqrt(x)

Square both sides of the equation to eliminate the square roots:

2x = x

Now, solve for x:

2x - x = 0 x = 0

However, you should check whether x = 0 is a valid solution by substituting it back into the original equation, as it might not satisfy the domain of the inverse trigonometric functions.

arcsin(sqrt(2*0)) = arcsin(0) = 0 arccos(sqrt(0)) = arccos(0) = π/2

Since arcsin(0) = 0 and arccos(0) = π/2, the solution x = 0 is valid. Therefore, the solution to the equation arcsin(sqrt(2x)) = arccos(sqrt(x)) is x = 0.

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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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