Perform each transformation. See Example 2.
Write sec x ...

Question

Perform each transformation. See Example 2.

Write sec x in terms of sin x.

Accepted Answer

Welcome back. I am so glad you're here. We are asked to write the given trigonometric function in terms of the cosine of X. Our given trigonometric function is the cosecant of X. Our answer choices are answer choice A plus or minus. And then we have a fraction in the numerator. It's the square R of one minus the cosine squared of X. And that's divided by in the denominator one plus the cosine squared of X answer choice B plus or minus again, a fraction in the numerator, the square root of one minus the cosine squared of X. And that's divided by in the denominator one minus cosine squared of X. Answer choice C plus or minus again, of fraction in the numerator, the square root of one plus the cosine squared of X divided by in the denominator one minus the cosine squared of X and answer choice D plus or minus in the numerator, the square root of one plus cosine squared X and that's divided by in the denominator one plus cosine squared X. All right. So cos can't getting that to cosine. Well, we recall from previous lessons that the reciprocal function for kant is signed. So the KC can of X is the same as one divided by the sine of X. So now we just need to get from sine to cosine. Well, let's think of some identities and equations that we know that contain both sine and cosine. We recall from previous lessons, the Pythagorean identity that sine squared of X plus the cosine squared of X is equal to one. Now let's solve this for the sine of X. So we'll subtract the cosine squared of X from both sides of the equation. Then on the left, we'll have the sine squared of X that's equal to on the right one minus the cosine squared of X. Then to get rid of that squared, we'll take the square rut of both sides. On the left, we have the sine of X great. That's what we're looking for. And that equals on the right, taking the square root will have plus or minus the square root of one minus cosine squared X. And now we can use substitution. So we know what the sine of X is equal to back in the denominator where we know the co sequent of X is equal to one divided by the sine of X. So we'll plug in what the sine of X is equal to. So one divided by the sine of X is the same as one divided by we have plus or minus. We can put that in front of the fraction. And then in the denominator, the sine of X is the same as the square root of one minus the cosine squared of X. But we don't wanna have a radical in our denominator. So we've got to rationalize this. We'll multiply both the numerator and the denominator by the square root of one minus the cosine squared of X because in effect, that's just multiplying it by one. And when we do that, when we simplify in the numerator, we now have the square root of one minus the cosine squared of X divided by, remember the whole thing is plus or minus. And then in the denominator multiplying one minus the cosine squared of X by itself will give us one minus the cosine squared X or excuse me, multiplying the square root of one minus cosine squared X by itself gives us one minus cosine squared of X. And there is our answer. We look at our answer choices for plus or minus square root of one minus cosine squared X all divided by one minus cosine squared X. And that matches with answer choice B well done. We'll catch you on the next one.

Perform each transformation. See Example 2.
Write sec x in terms of sin x.

Key Concepts

Reciprocal Trigonometric Functions

Pythagorean Identity

Domain and Sign Considerations

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