Find each exact function value. See Example 2.
sec 23π/6...

Question

Find each exact function value. See Example 2.

sec 23π/6

Accepted Answer

Welcome back. I am so glad you're here. We are asked to determine the exact value of the given function. Our given function is the C can of 47 pi divided by six. Our answer choices are answer choice. A negative square root three divided by two answer choice B negative two square root three divided by three. Answer choice C two square root three divided by three and answer choice D square at three divided by two. All right. So 47 pi divided by six, that angle is not on the unit circle, but we can try to find out if it has a co terminal angle on the unit circle. So we've got a big number here. Let's start with 47 pi divided by six and we will subtract one full rotation until we get somewhere between zero and chew pie. So 47 pi divided by six minus two pi equals 35 pi divided by six. Not there yet. We'll subtract another two pie. 35 pi divided by six minus two. Pi is 23 pi divided by six. Not there yet. We'll subtract another two pie. 23 pi divided by six minus two. Pi gives us 11 pi divided by six. And that one is on the unit circle. So 47 pi divided by six is co terminal with 11 pi divided by six. Now let's see where that was on the unit circle. We recall from previous lessons and you can take a look at the image of the unit circle in your textbook that the unit circle gives us a vertical y axis, a horizontal X axis. They come together at the origin in the middle, the unit circle over that has every point on it. One unit away from the origin, the one that I drew doesn't. But you can imagine that it does. And for 11 pi divided by six, if we begin at the term or excuse me at the initial side along the positive part of the X axis, we had counterclockwise, we're gonna go all the way over into the fourth quadrant and we are going to be closer to the positive part of the X axis than the negative part of the Y axis. There's our 11 pi divided by six, 11 pi divided by six is equal to 330 degrees. And from that image in the textbook, we can see that the X coordinate at 11 pi divided by six is equal to square at three divided by two and the Y coordinate is negative one half. Why is this important? Well, we recall from previous lessons that the sea can't of an angle on the unit circle. So we'll call that S the C can of S is equal to one divided by X. And here our X is square root three divided by two. So this is one divided by square root three divided by two. When we have division within division, when there's a fraction within a fraction, I like to rewrite this with the division symbol. So rewriting it here, one divided by square root three, divided by two. Because that reminds me that division with fractions is the same as multiplying by the reciprocal of the second fraction. So this is one multiplied by two divided by square root three. And so this is just two divided by square at three. But we do want to rationalize this denominator. So we'll multiply both the numerator and the denominator by the square root of three. And in the numerator, we've got two square root three divided by and the denominator, the square root of three multiplied by itself is just three. And there is our C can of 47 pi divided by six. We look at our answer choices and this matches with answer choice. C well done. We'll catch you on the next one.

Find each exact function value. See Example 2.
sec 23π/6

Key Concepts

Secant Function

Unit Circle

Angle Measurement in Radians

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