Problem 30 - A Graphical Approach to Precalculus with Limits

Question

Find the exact value of each real number \(y .\) Do not use a calculator. $$y=\csc ^{-1} \sqrt{2}$$

Step-by-Step Solution

Answer

The exact value of \( y \) is \( \frac{\pi}{4} \).

Step 1: Understand the inverse cosecant function

The function \( \ ext{csc}^{-1}(x) \) is the inverse of the cosecant function. The range of \( \ ext{csc}^{-1}(x) \) is \( [-\frac{\pi}{2}, 0) \cup (0, \frac{\pi}{2}] \). This means we are looking for an angle \( y \) such that \( \csc(y) = \sqrt{2} \).

Step 2: Write the cosecant in terms of sine

Recall that \( \csc(y) = \frac{1}{\sin(y)} \). Therefore, \( \csc(y) = \sqrt{2} \) can be rewritten as \( \frac{1}{\sin(y)} = \sqrt{2} \), or equivalently \( \sin(y) = \frac{1}{\sqrt{2}} \).

Step 3: Determine the reference angle

The value \( \sin(y) = \frac{1}{\sqrt{2}} \) corresponds to the reference angle \( y = \frac{\pi}{4} \) since \( \sin\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} \).

Step 4: Choose the angle in the correct range

We need \( y \) to lie in the range of the inverse cosecant function, which is \([\frac{-\pi}{2}, 0) \cup (0, \frac{\pi}{2}] \). The angle \( y = \frac{\pi}{4} \) is in this range. Therefore, the exact value of \( y \) is \( \frac{\pi}{4} \).

Key Concepts

Cosecant Function

The cosecant function, denoted as \( \csc(x) \), is one of the six fundamental trigonometric functions, and it is the reciprocal of the sine function. This means if you have a sine value, you can find the cosecant by taking the reciprocal:

• \( \csc(x) = \frac{1}{\sin(x)} \)

In trigonometry, cosecant is particularly useful for finding the ratio of the hypotenuse to the opposite side in a right triangle, just like sine gives the opposite over hypotenuse ratio. The range of the function \( \csc(x) \) is important, especially when dealing with its inverse function, \( \csc^{-1}(x) \). The inverse cosecant function is the function that "undoes" the \( \csc \) function, finding an angle when the cosecant is known. Its range is \([-\frac{\pi}{2}, 0) \cup (0, \frac{\pi}{2}]\), which allows us to find angles in the one cycle between these intervals.

Sine Function

The sine function, noted as \( \sin(x) \), is a primary trigonometric function that calculates the ratio of the length of the side of a right triangle opposite to an angle to the length of the triangle's hypotenuse. It is a key concept in understanding other trigonometric functions, including the cosecant.The values of the sine function range from \(-1\) to \(1\), which stick to easily finding specific angles from known sine values because these characteristics give us a bounded function. In the unit circle, \( \sin(x) \) values help find specific angles due to the circular symmetry:

• \( \sin(\frac{\pi}{4}) = \frac{1}{\sqrt{2}} \) – an important relation when finding specific angles where sine is used in expressions like the square root of two fractions.

Understanding the sine function is crucial to rewriting \( \csc(y) = \sqrt{2} \) as \( \sin(y) = \frac{1}{\sqrt{2}} \) in solving inverse trigonometric exercises.

Reference Angle

A reference angle is the smallest angle the terminal side of a given angle makes with the x-axis. It's particularly handy in trigonometry, as it helps in finding trigonometric values for any angle.Whenever you are given a trigonometric function's value, you can first find the corresponding reference angle, allowing you to determine the original angle. For instance, if \( \sin(y) = \frac{1}{\sqrt{2}} \), you can immediately associate it with the reference angle \( \frac{\pi}{4} \), because:

• The angle \( \frac{\pi}{4} \) yields \( \sin(\frac{\pi}{4}) = \frac{1}{\sqrt{2}} \), thus telling us about this common scenario.

By determining the reference angle, you oftentimes need to check the quadrant to make sure your final angle lies within the specified range, like choosing \( y = \frac{\pi}{4} \) since it falls into the set range of \([\frac{-\pi}{2}, 0) \cup (0, \frac{\pi}{2}] \). Reference angles simplify trigonometric exploration, letting you navigate through exact value calculations with ease.