Problem 8 - Algebra and Trigonometry

Question

Find the exact value of the trigonometric function at the given real number. $$ \begin{array}{llll}{\text { (a) } \sin \frac{3 \pi}{4}} & {\text { (b) } \sin \frac{5 \pi}{4}} & {\text { (c) } \sin \frac{7 \pi}{4}}\end{array} $$

Step-by-Step Solution

Answer

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

Step 1: Understand the Unit Circle

The unit circle is centered at the origin (0,0) on the coordinate plane and has a radius of 1. It is used to define trigonometric functions for all angles. The angle \( \theta \) can be represented by a point \((x, y)\) on the circle, where \( x = \cos(\theta) \) and \( y = \sin(\theta) \).

Step 2: Finding \( \sin \frac{3\pi}{4} \)

The angle \( \frac{3\pi}{4} \) is in the second quadrant. In the second quadrant, sine is positive. The reference angle is \( \frac{3\pi}{4} - \pi/2 = \frac{\pi}{4} \). The sine of \( \frac{\pi}{4} \) is \( \frac{\sqrt{2}}{2} \). Therefore, \( \sin \frac{3\pi}{4} = \frac{\sqrt{2}}{2} \).

Step 3: Finding \( \sin \frac{5\pi}{4} \)

The angle \( \frac{5\pi}{4} \) is in the third quadrant. In the third quadrant, sine is negative. The reference angle is \( \frac{5\pi}{4} - \pi = \frac{\pi}{4} \). The sine of \( \frac{\pi}{4} \) is \( \frac{\sqrt{2}}{2} \), so \( \sin \frac{5\pi}{4} = -\frac{\sqrt{2}}{2} \).

Step 4: Finding \( \sin \frac{7\pi}{4} \)

The angle \( \frac{7\pi}{4} \) is in the fourth quadrant. In the fourth quadrant, sine is negative. The reference angle is \( 2\pi - \frac{7\pi}{4} = \frac{\pi}{4} \). The sine of \( \frac{\pi}{4} \) is \( \frac{\sqrt{2}}{2} \), so \( \sin \frac{7\pi}{4} = -\frac{\sqrt{2}}{2} \).

Key Concepts

Trigonometric Functions

Trigonometric functions are mathematical functions that relate the angles of a triangle to the lengths of its sides. The most common ones are sine (\( \sin \theta \), cosine (\( \cos \theta \)), and tangent (\( \tan \theta \)). They are essential in various fields such as physics, engineering, and even music technology.
The sine function, in particular, uses the opposite side and the hypotenuse of a right-angled triangle, and it is commonly used in wave analysis and simple harmonic motion. In the unit circle, the sine of an angle is represented by the y-coordinate of a point on the circle.

• Sine function: measures the height from the x-axis
• Cosine function: measures the distance from the origin to the x-axis
• Tangent function: ratio of sine to cosine

Understanding these basics will help you compute angles and distances in different applications.

Reference Angles

A reference angle is the smallest angle made by the terminal side of a given angle and the x-axis. Finding this angle helps simplify the computation of trigonometric functions.
Reference angles are always acute (less than 90 degrees) and are used to determine the sine, cosine, and other trigonometric values for angles in different quadrants.

• For angle \( \theta \) in quadrant II, reference angle = \( \pi - \theta \)
• For angle \( \theta \) in quadrant III, reference angle = \( \theta - \pi \)
• For angle \( \theta \) in quadrant IV, reference angle = \( 2\pi - \theta \)

By knowing the reference angle and the quadrant, you can simply adjust the sign as needed to find the exact function value. It's a powerful technique to master for anyone working with angles.

Quadrants

The coordinate plane is divided into four sections known as quadrants, each governed by different signs and rules. Understanding these quadrants is crucial for determining the sign of trigonometric functions.

• Quadrant I: Both x and y are positive. All trigonometric functions are positive.
• Quadrant II: x is negative, y is positive. Sine is positive.
• Quadrant III: Both x and y are negative. Tangent is positive.
• Quadrant IV: x is positive, y is negative. Cosine is positive.

When solving problems, knowing which quadrant your angle lies in helps determine whether the sine, cosine, or tangent of an angle will be positive or negative.
For example, understanding that \( \frac{3\pi}{4} \) lies in Quadrant II helps you know that \( \sin \frac{3\pi}{4} \) will be positive, while \( \cos \frac{3\pi}{4} \) will be negative.
Mastering these quadrant rules is a fundamental step in navigating trigonometric problems with confidence.