Question

Use the even-odd properties to find the exact value of each expression. Do not use a calculator. $$ \csc \left(-\frac{\pi}{4}\right) $$

Short answer

-\sqrt{2}

Step-by-step solution

Identify the function

The function to evaluate is \(\csc(-\frac{\pi}{4})\). \( \csc \) is the cosecant function, which is the reciprocal of the sine function: \(\csc(x) = \frac{1}{\sin(x)}\).

Use the Even-Odd Property

Use the even-odd property of the sine function. The sine function is odd, meaning that \( \sin(-x) = -\sin(x) \). Thus, \(\sin(-\frac{\pi}{4}) = -\sin(\frac{\pi}{4})n\).

Find the Sine Value

Recall that \(\sin(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}\). Using the odd property of the sine function: \(\sin(-\frac{\pi}{4}) = -\frac{\sqrt{2}}{2}\).

Find the Reciprocal

To find the cosecant value, take the reciprocal of the sine value found in Step 3. Thus, \(\csc(-\frac{\pi}{4}) \) is \(\frac{1}{-\sin(\frac{\pi}{4})} = \frac{1}{-\frac{\sqrt{2}}{2}}\).

Simplify

Simplify the reciprocal: \(\frac{1}{-\frac{\sqrt{2}}{2}} = -\frac{2}{\sqrt{2}} = -\sqrt{2}\). Thus, the exact value of \(\csc(-\frac{\pi}{4})\) is \(-\sqrt{2}\).

Key concepts

even-odd properties

Understanding even-odd properties in trigonometry is essential. These properties help simplify expressions and solve problems without a calculator.

The sine function, denoted as \(\text{sin}(x)\), is an **odd function**. This means that it has the property \[ \text{sin}(-x) = -\text{sin}(x) \].

On the other hand, even functions follow the property \[ f(-x) = f(x) \].

By recognizing these properties, it is easier to work with negative angles in trigonometric functions. For the given problem, this property helps us transform a potentially complicated negative angle into something simpler to handle. So, instead of dealing with \(-\pi/4\), we can transform this as \(-\text{sin}(\pi/4)\).

cosecant function

The cosecant function is one of the fundamental trigonometric functions. Denoted as \( \text{csc}(x) \), it is defined as the reciprocal of the sine function.

Mathematically, this relationship can be expressed as: \[\text{csc}(x) = \frac{1}{\text{sin}(x)} \].

This means to find the value of cosecant at any angle, you first need to know the sine value of that angle. This reciprocal property often makes it necessary to understand and transform sine values efficiently.

In the current exercise, because we have \( \text{csc}(-\pi/4) \), the steps would involve first calculating \( \text{sin}(-\pi/4) \) and then finding its reciprocal.

sine function

The sine function is one of the most fundamental functions in trigonometry. It relates the angle of a right triangle to the ratio of the length of the side opposite the angle to the hypotenuse.

For example, \(\text{sin}(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \).

In the context of unit circle definitions, \( \text{sin}(x) \) gives the y-coordinate of the point on the circle corresponding to angle \( x \).

To calculate the sine value for common angles like \( \pi/4 \), remember: \[ \text{sin}(\pi/4) = \frac{\sqrt{2}}{2} \].

Using the odd property of the sine function, \( \text{sin}(-\pi/4) = -\frac{\sqrt{2}}{2} \). This symmetrical property means whether the angle is positive or negative, you can calculate the sine value easily with just a sign change.