Find the exact value of each real number y. Do not use a calcu...

Question

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

y = sin⁻¹ √2/2

Accepted Answer

Hello. Today we are going to be determining the exact value of the given inverse trigonometric function without using a calculator. So we are told to solve for Y is equal to sine inverse of zero plus sine inverse of negative one. So how can we start this process? Well, in order to evaluate sine inverse of zero and sine inverse of negative one, we must understand the domains of the function sine inverse. Now the domain of the function sine inverse is given to us as negative pi divided by two comma pi divided by two. Now, on the unit circle, what this means is that the domain of sine inverse is going to be the right half of the unit circle. So whenever we want to find the value of a sin inverse function, that value will only be an angle measure within this domain. Let's go ahead and start by solving for sign inverse of zero. So recall that whenever we solve for an inverse trigonometric function, the output is always going to be some angle theta. So we can take sine inverse of zero and set it equal oops sorry sine inverse of zero and set it equal to some angle theta. If we were to take the sine function of both sides of this equation, we can rewrite the equation to be zero is equal to sine theta. And if we reorder the terms, we can rewrite this equation to be sine theta is equal to zero. So we need to ask ourselves, what angle can we plug into the sign that will give us an output of zero? And we want that angle to be within the domain from negative pi divided by two to pi divided by two. Well, if we stay with respect to this domain, the only value where sine is equal to zero is going to be on the angle zero. So what this means is that the value s inverse of zero is equal to zero. That is going to be the value of the first trigger trigonometric inverse function. Now let's go ahead and evaluates inverse of negative one. Now, just like before we're going to set sine inverse of negative one equal to some angle theta, if we take the sine function of both sides of this equation, we will have a negative one is equal to sine of data. And reordering the terms of the equation will allow us to rewrite the equation as sine theta is equal to negative one. Now, just like before we want to find the value of data that we can plug into the sine function that will give us an output of negative one. And that value or angle has to be within the domain of negative pi divided by two to pi divided by two. Well, for us sine theta equal to negative one, that angle lies right on the domain at the value negative pi divided by two. So what this means is that the inverse function sine inverse of negative one is equal to negative pi divided by two. That is going to be the output that we are looking for now that we have both of the values we can solve for the value of Y. So again, Y is equal to sine inverse of zero plus sine inverse of negative one. Since we solved for both of the inverse functions, we can rewrite the equation to be Y is equal to zero minus pi divided by two. And this will allow us to simplify the equation to be Y is equal to negative pi divided by two. This is going to be the exact value of the inverse trigonometric function without using a calculator. And with that being said, the answer to this problem is going to be D 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.

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

Key Concepts

Inverse Trigonometric Functions

Unit Circle

Special Angles

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