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 = sin⁻¹ (―1)

Accepted Answer

Hello, everyone. We are asked to determine the exact value of the given inverse trigonometric function. Without using a calculator, we are given Y equals the arc sign of negative radical two divided by two. Our answer choices are A Y equals pi divided by four by equals negative pi divided by three cy equals negative pi divided by four dy equals negative pi divided by six. First, since we were asked to do this without using a calculator, you wanna make sure you have a unit circle on hand. If you do not, you might want to pause and grab one. Now, next, thinking about the arc sine function, I know we have a domain that is between negative one and one including the end points and a range that is between negative pi divided by two and pi divided by two. Again, including the end points. Given this range, our arc sign must fall in either quadrant one or quadrant four. Since this arc sine is negative and sine is negative in quadrant four. I know I'm looking in the fourth quadrant of my unit circle. So when I find negative radical two divided by two, the angle measure that is with, that is seven pi divided by four. Well, thinking about my unit circle and just sketching a little coordinate plane over here, typically, our angles are counted in a counterclockwise motion and that's what makes them positive. And that is how we get a positive seven pi divided by four. However, if I go in the opposite direction, I go clockwise, we would have negative angles. So to find our final angle here, I'm gonna do seven pi divided by four minus two pi since that is the length of the entire circle, so that would be seven pi divided by four minus eight pi divided by four, which leaves us with negative pi divided by four. So that would be our co angle and our answer. So negative pi divided by four is answer choice C and that is it have a nice day.

Find the exact value of each real number y if it exists. Do not use a calculator.
y = sin⁻¹ (―1)

Key Concepts

Inverse Sine Function (sin⁻¹ or arcsin)

Domain and Range of the Sine Function

Exact Values of Sine at Special Angles

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