Problem 21 - Algebra and Trigonometry

Question

Find the exact value of the trigonometric function at the given real number. (a) \(\sin \frac{5 \pi}{4} \quad\) (b) \(\csc \frac{\pi}{2} \quad\) (c) \(\csc \frac{3 \pi}{2}\)

Step-by-Step Solution

Answer

(a) \(-\frac{\sqrt{2}}{2}\), (b) 1, (c) -1

Step 1: Identify and Use Reference Angle (a)

The angle \( \frac{5\pi}{4} \) is in the third quadrant. Its reference angle is found by subtracting \( \pi \) (or \( 180^\circ \)) from \( \frac{5\pi}{4} \). This gives \( \frac{5\pi}{4} - \pi = \frac{\pi}{4} \). Since sine is negative in the third quadrant, \( \sin \frac{5\pi}{4} = -\sin \frac{\pi}{4} = -\frac{\sqrt{2}}{2} \).

Step 2: Evaluate Cosecant Function (b)

To find \( \csc \frac{\pi}{2} \), we first find the sine of \( \frac{\pi}{2} \). The sine of \( \frac{\pi}{2} \) is 1. By definition, the cosecant is the reciprocal of the sine function. Therefore, \( \csc \frac{\pi}{2} = \frac{1}{\sin \frac{\pi}{2}} = \frac{1}{1} = 1 \).

Step 3: Evaluate Cosecant Function (c)

For \( \csc \frac{3\pi}{2} \), find \( \sin \frac{3\pi}{2} \). The sine of \( \frac{3\pi}{2} \) is -1. Then, \( \csc \frac{3\pi}{2} = \frac{1}{\sin \frac{3\pi}{2}} = \frac{1}{-1} = -1 \).

Key Concepts

Sine Function

The sine function is one of the fundamental trigonometric functions. It is often denoted by "sin" and is used to relate the angle of a right-angled triangle to the ratio of the length of the opposite side to the hypotenuse. This relationship is given by the formula:

• For an angle \( \theta \), \( \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \)

The sine function is periodic, repeating every \( 2\pi \) radians or 360 degrees. This means that it has the same value at interval lengths of \( 2\pi \) radians.
Additionally, the sine function exhibits particular behavior in each of the four quadrants of the unit circle:

• In the first quadrant \((0, \pi/2)\), sine values are positive.
• In the second quadrant \((\pi/2, \pi)\), sine values remain positive.
• In the third quadrant \((\pi, 3\pi/2)\), sine values are negative.
• In the fourth quadrant \((3\pi/2, 2\pi)\), sine values continue to be negative.

These properties help determine the sign of the sine function when evaluating angles beyond the first circle. As seen in the original exercise, for \( \sin \frac{5\pi}{4} \), the function takes on a negative value, which tells us the angle is in the third quadrant.

Reference Angle

A reference angle is the acute angle that a given angle makes with the x-axis. Calculating the reference angle is essential in trigonometry because it helps simplify the evaluation of trigonometric functions for angles beyond one full cycle, or beyond \( 2\pi \) or 360°.
To find the reference angle:

• For angles in the first quadrant, the reference angle is the angle itself.
• For angles in the second quadrant, subtract the angle from \( \pi \).
• For angles in the third quadrant, subtract \( \pi \) from the angle.
• For angles in the fourth quadrant, subtract the angle from \( 2\pi \).

In our exercise, with \( \frac{5\pi}{4} \), you find the reference angle by noting it is in the third quadrant and performing the calculation: \( \frac{5\pi}{4} - \pi = \frac{\pi}{4} \). The reference angle \( \frac{\pi}{4} \) corresponds to 45 degrees in standard angle measure, which is helpful for determining trigonometric values.

Cosecant Function

The cosecant function, represented as "csc", is the reciprocal of the sine function. It is less commonly used than the sine, cosine, or tangent functions, but it is still an important part of trigonometry.
The formula for the cosecant function is:

• \( \csc(\theta) = \frac{1}{\sin(\theta)} \)

This reciprocal relationship means that cosecant is undefined wherever sine equals zero, as division by zero is undefined.
Exploring further using the original exercise:

• For \( \csc \frac{\pi}{2} \), sine is 1, so \( \csc \frac{\pi}{2} = 1 \).
• For \( \csc \frac{3\pi}{2} \), sine is -1, so \( \csc \frac{3\pi}{2} = -1 \).

Knowing the sine values at these critical angles \((\pi/2, 3\pi/2)\) is essential as they frequently appear in trigonometry problems. Reminder: The cosecant function will have the same sign as the sine function since it is its reciprocal.