Find the exact value of each real number y if it exists. Do no...

Question

Find the exact value of each real number y if it exists. Do not use a calculator.

y = csc⁻¹ √2/2

Accepted Answer

Welcome back. I am so glad you're here. We are asked to determine the exact value of the given inverse trigonometric function without using a calculator. Our given function is Y equals the inverse cosecant of the square of three divided by two. Our answer choices are answer choice A Y equals pi divided by three answer choice by equals negative pi divided by three answer choice cy equals pi divided by six. And answer choice D does not exist. All right. The first question we need to ask ourselves is does the square root of three divided by two fall within the domain of the inverse co he can't function. And I'm going to draw a few quick sketches of some functions in order to help us figure that out if you need a review on graphing trigonometric functions, definitely please check out previous lesson videos. So we've got a vertical Y axis and a horizontal X axis. They come together at the origin cos can is the reciprocal function of the sine function. So I'm going to start off by graphing assign function. The sine function has its first point of note at the origin at 00 and then it begins curving and extends up to the point pi divided by 21. Now, normally we would follow that sine curve, it would head back down to the point pi zero. But we need all of these functions to be 1 to 1 function. So the Y values cannot repeat. So instead of continuing to the right along the X axis, I'm going to back it up before the origin, the first point of note would be at negative pi divided by two negative one. And so that is the domain that we're using for our sine function. So I'm going to smoothly connect these from negative pi divided by two negative one curves up passes through the origin at 00, continues curving up at the point pi divided by 21. Now that's ours function but that helps us graph our cos can't function, the cos can't function begins at negative pi divided by two negative one. And then instead of curving up, it curves down toward negative infinity for the Y values along the vertical Asymptote at X equals zero or the Y axis to the right of that vertical Asymptote. It's heading up toward positive infinity for the Y values. Then it curves down and intersects at the point pi divided by 21. There's our Kamp function, but we're trying to figure out the inverse kosic camp function. Well, the inverse Kamp function has all of its Y values are the Koi cans functions X values and all of its X values are the Koi cans functions Y values. So we're just swapping those, which means that to figure out the domain of our inverse Koi can function, we just need to figure out the range of our regular KC can function. So what is the range of our Koi Camp function? The range we can see that it starts down at negative infinity, then heads up to negative one inclusive inclusive of that. So we'll put a bracket that's in union with and then it doesn't start up again for the Y values until we'll put a bracket positive one and then continues all the way up to infinity. So that's the range of our Koi Camp function. It's going to be the domain of our inverse Koi Camp function from negative infinity to negative one inclusive of that in union with inclusive of positive one all the way up to infinity. Now does the square root of three divided by two fall within that domain? Well, if you put square root of three divided by two into a calculator, I know it says don't use a calculator. But if you don't know what that is off the top of your head, that's going to be 0.866. And then it continues. But we can see that this falls in between negative one and one. It is outside of our domain. So this Y equals the inverse Kan of the square root of three divided by two does not exist. We look at our answer choices and this matches with answer choice. D well done. We'll catch you on the next one.

Find the exact value of each real number y if it exists. Do not use a calculator.
y = csc⁻¹ √2/2

Key Concepts

Inverse Cosecant Function (csc⁻¹)

Reciprocal Relationship Between Sine and Cosecant

Unit Circle Values and Exact Trigonometric Ratios

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