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)

Accepted Answer

Hello, everyone. We're asked to determine the exact value of the given inverse trigonometric function. Without using a calculator, we're given Y equals the inverse kant of negative two radical three divided by three. Our answer choices are A Y equals pi divided by three by equals negative pi divided by three cy equals negative pi divided by six dy equals pi divided by six. So since we're not using a calculator, we will be using a unit circle first to make sure we can do this. I need to consider the domain of our inverse K can function and that is from negative infinity to negative one including negative one union with one to positive infinity. So does our value here. Our negative two radical three divided by three, we call X fall into our domain. Yes, it does. So that means we can continue. So we're gonna consider the range of this function which is from negative pi divided by two including pi divided by two negative pi divided by 2 to 0 union with zero two pi divided by two including that end point. So this means we are considering quadrants one and four. So it's either quadrant one or quadrant four that this value lies in. So since the inverse KC can't of negative two radical three divided by three equals Y I could rewrite this as the cosecant of why equals negative two radical three divided by three. Since that's not on my unit circle, I recall my reciprocal identity and that the kant of Y would equal one divided by the sine of Y. So this means that I can take the reciprocal of negative two radical three divided by three and find the sign of it. So this would be the sign of Y is going to equal the negative square root of three divided by two. And when I consider that on my unit circle, why must equal five pi divided by three? So it's five pi divided by three. If I consider my coordinate plane here and starting at our standard positioning and going counterclockwise. However, if we're considering going clockwise, it would be a negative. So this would actually turn out to be negative pi divided by three. So our answer is that Y equals negative pi divided by three which is answer choice B have a nice day.

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

Key Concepts

Cosecant Function

Inverse Trigonometric Functions

Domain and Range of Inverse Functions

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