Write each trigonometric expression as an algebraic expression...

Question

Write each trigonometric expression as an algebraic expression in u, for u > 0.

cos (arcsin u)

Accepted Answer

Hello. Today we are going to be rewriting the given trigonometric expression as an algebraic expression with the terms X, we will also be assuming that the value of X is greater than zero. So we are given cosine of sine inverse of four X. The first thing we want to do is we want to simplify this trigonometric expression. Now recall that whenever we solve for trigonometric function, whether it be a standard function or an inverse function, its output is always going to be some angle theta. What this means is that we can take the inside angle sign inverse of four X and set it equal to some angle theta. This will allow us to simplify the trigonometric expression to be cosine of theta. Now, the problem here is that we don't have any way to further simplify cosine of data. And we don't have any way to currently write it as an algebraic expression. So how can we continue from the step? Well, we remember how we said sine inverse of four X equal to theta. Well, if we were to take sine of both sides of this equation, we can rewrite the equation to be four X is equal to sin of theta. And if we reorder the terms in the equation, we can set sine theta equal to four X. Now, you may be wondering why I just solved for sign of data. The reason for this is because we're going to use the value sine theta equal to four X to create a right triangle. We're going to set the angle theta to the far left corner of the right triangle. And since we have the value of sine data equal to four X, we can use this value to label the sides of the right triangle and use those sides to rewrite cosine data as an algebraic expression recall that the trigonometric value for sine theta is equal to the opposite side with respect to theta divided by the hypotenuse of the right triangle. This means that sine theta is equal to a fraction. Now, currently, we have sine theta is equal to four X, but we can represent four X as a fraction by setting it divided by one. So that means the numerator four X is going to be the opposite side with respect to theta. So we're going to label the opposite side as four X, the denominator which is one is going to the B be the hypotenuse of the right triangle. So the hypotenuse will be one, we are only missing one side to this right triangle. And we're going to label that side A let's go ahead and solve for this value A we can solve for the value of A by using the Pythagorean theorem. The Pythagorean theorem will allow us to write one squared is equal to A squared plus four X squared. Simplifying this equation will give us one is equal to a squared plus 16 X squared. Since we want to solve for a, we're going to subtract 16 X squared to both sides of the equation. This will leave us with A squared is equal to one minus 16 X squared. And if we take the square root of both sides of this equation, the value of A will equal to the square root of one minus 16 X squared. So since we just solve for the value of A, we can re label the adjacent side of the angle theta to be the square root of one minus 16 X squared. Now that we have all three values of the sides of the right triangle, we can use this to rewrite cosine theta as an algebraic expression. Recall that the trigonometric value for cosine of data is equal to the adjacent side of the right triangle divided by the hypotenuse of the right triangle. We just solved for the adjacent side. The adjacent side is the square root of one minus 16 X squared. And we know the value of the Hypo news. The Hypo news is one that means we can rewrite cosine data to be the square root of one minus 16 X squared divided by one. And since the numerator to this expression is just one, we can further simplify the expression to be the square root of one minus 16 X squared. This is going to be the simplest form of the algebraic expression. And with that being said, the answer to this problem is going to be B so I hope this video helps you in understanding how to approach this problem. And I'll go ahead and see you all in the next video.

Write each trigonometric expression as an algebraic expression in u, for u > 0.
cos (arcsin u)

Key Concepts

Inverse Trigonometric Functions

Pythagorean Identity

Trigonometric Ratios

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