Problem 28 - Algebra and Trigonometry

Question

9–32 Find the exact value of the trigonometric function. $$\csc \frac{5 \pi}{4}$$

Step-by-Step Solution

Answer

The exact value is \( -\sqrt{2} \).

Step 1: Understand the Problem

The problem requires us to find the exact value of the cosecant function \( \csc \left( \frac{5 \pi}{4} \right) \). Recall that \( \csc(\theta) = \frac{1}{\sin(\theta)} \). Thus, we need to find \( \sin \left( \frac{5 \pi}{4} \right) \) first.

Step 2: Determine the Angle's Quadrant

The angle \( \frac{5 \pi}{4} \) is in radians and equivalent to \( 225^\circ \). This angle is greater than \( 180^\circ \) but less than \( 270^\circ \), placing it in the third quadrant of the unit circle where the sine value is negative.

Step 3: Find Reference Angle

The reference angle for \( \frac{5 \pi}{4} \) is obtained by subtracting \( 180^\circ \) (or \( \pi \) radians) from it, which yields \( 45^\circ \) or \( \frac{\pi}{4} \).

Step 4: Calculate the Sine Value

In the third quadrant, the sine of \( \frac{5\pi}{4} \) is negative. The sine of a \( 45^\circ \) angle (reference angle) is \( \frac{\sqrt{2}}{2} \). Thus, \( \sin \left( \frac{5\pi}{4} \right) = -\frac{\sqrt{2}}{2} \).

Step 5: Calculate the Cosecant Value

Use the identity \( \csc(\theta) = \frac{1}{\sin(\theta)} \) to find \( \csc \left( \frac{5\pi}{4} \right) \). Since \( \sin \left( \frac{5\pi}{4} \right) = -\frac{\sqrt{2}}{2} \), it follows that \( \csc \left( \frac{5\pi}{4} \right) = \frac{1}{-\frac{\sqrt{2}}{2}} = -\frac{2}{\sqrt{2}} = -\sqrt{2} \).

Key Concepts

Trigonometric Functions

Trigonometric functions are fundamental in mathematics. They are used to describe relationships in triangles, especially right-angled ones. - **Common Functions**: The primary trigonometric functions are sine (\(\sin\)), cosine (\(\cos\)), tangent (\(\tan\)), cosecant (\(\csc\)), secant (\(\sec\)), and cotangent (\(\cot\)).- **Defining Relationships**: These functions are defined using the ratio of sides in a right triangle. For example, the cosecant function is \(\csc(\theta) = \frac{1}{\sin(\theta)}\).Every trigonometric function describes a unique relationship between an angle and the proportion of any two sides of a right-angled triangle. They also extend to periodic functions on the coordinate plane, helping to solve problems involving oscillations, waves, circles, and rotations. When dealing with trigonometric identities, it's crucial to remember these reciprocal relationships to solve equations involving angles effectively.

Unit Circle

The unit circle is an essential concept in understanding trigonometric functions beyond triangles. It is a circle with a radius of 1 unit and is centered at the origin of a coordinate plane.- **Purpose**: The unit circle allows us to define trigonometric functions for any angle, positive or negative, and to easily visualize and memorize the function values.- **Quadrants**: The unit circle is divided into four quadrants: 1. Quadrant I: All trigonometric functions are positive. 2. Quadrant II: Sine is positive, while cosine and tangent are negative. 3. Quadrant III: Tangent is positive, while sine and cosine are negative. 4. Quadrant IV: Cosine is positive, while sine and tangent are negative.For the angle \(\frac{5\pi}{4}\), we know it lies in Quadrant III, where sine values, and thus \(\csc\), are negative. Understanding the placement of angles on the unit circle helps in accurately determining the sign and value of trigonometric functions.

Reference Angle

When working with angles and trigonometric functions, the reference angle is a vital concept. It is the smallest angle that the terminal side of the given angle makes with the x-axis. - **Importance**: Knowing the reference angle helps in finding the trigonometric function values of angles not located in the first quadrant by using the values from the first quadrant but considering the correct sign based on the quadrant location.- **Calculation**: For example, the reference angle for \(\frac{5\pi}{4}\) (which corresponds to \(225^\circ\)) is \(\frac{\pi}{4}\) (or \(45^\circ\)) because \(225^\circ - 180^\circ = 45^\circ\).By using the reference angle, we can easily determine that the sine of \(225^\circ\) has the same magnitude as the sine of \(45^\circ\), which is \(\frac{\sqrt{2}}{2}\), but with a negative sign in Quadrant III where \(\frac{5\pi}{4}\) resides.