Problem 19 - Algebra and Trigonometry

Question

\(17-24\) n Solve the given equation, and list six specific solutions. $$ \sin \theta=\frac{\sqrt{2}}{2} $$

Step-by-Step Solution

Answer

Solutions are \( \frac{\pi}{4}, \frac{3\pi}{4}, \frac{9\pi}{4}, \frac{11\pi}{4}, \frac{17\pi}{4}, \frac{19\pi}{4} \).

Step 1: Recognize the basic solutions

The equation \( \sin \theta = \frac{\sqrt{2}}{2} \) represents a standard sine value for which we know basic angle solutions. These angles are \( \theta = \frac{\pi}{4} \) and \( \theta = \frac{3\pi}{4} \) in the interval \([0, 2\pi)\).

Step 2: Identify the periodic nature of sine

Since sine is a periodic function with a period of \( 2\pi \), every solution \( \theta \) can be expressed as \( \theta = \theta_0 + 2k\pi \), where \( \theta_0 \) is a specific solution and \( k \) is an integer.

Step 3: Generate further solutions

Starting with the basic solutions: \( \theta = \frac{\pi}{4} + 2k\pi \) and \( \theta = \frac{3\pi}{4} + 2k\pi \). Plugging in values for \( k \), starting from \( k = 0 \), we have specific solutions. These are \( \theta = \frac{\pi}{4}, \frac{3\pi}{4}, \frac{9\pi}{4}, \frac{11\pi}{4}, \frac{17\pi}{4}, \frac{19\pi}{4} \).

Key Concepts

Sine Function

The sine function, represented as \( \sin \theta \), is one of the primary trigonometric functions. It relates a right triangle's angle with the ratio of the length of the opposite side to the hypotenuse. One of the most important properties of the sine function is that it only takes values between -1 and 1, inclusive. This trait makes it very useful in various fields like engineering and physics, especially when modeling wave and oscillatory phenomena.

When we look at the sine function on a graph, it forms a smooth, periodic wave. The range from -1 to 1 repeats over its domain, giving the sine wave its undulating shape. Key angles, like \( \theta = \frac{\pi}{4} \), where \( \sin \theta = \frac{\sqrt{2}}{2} \), are frequently encountered in problems involving trigonometric equations. Memorizing these standard angles can greatly speed up the problem-solving process.

Periodic Function

A periodic function is a function that repeats its values at regular intervals. The sine function is a classic example, repeating every cycle of \( 2\pi \) radians. This means that if \( \sin \theta = x \), then \( \sin(\theta + 2k\pi ) = x \) for any integer \( k \).

Understanding the periodic nature of the sine function helps in solving trigonometric equations, as seen in the exercise. By knowing that \( \theta = \frac{\pi}{4} \) and \( \theta = \frac{3\pi}{4} \) are solutions within one period, we can extend these solutions by adding multiples of \( 2\pi \) to find additional solutions. This property is essential when the equation asks for solutions over a wide interval or a specific number of solutions.

Solutions in Radians

When dealing with trigonometric equations, it's common to find solutions in terms of radians instead of degrees. Radians provide a more natural measure of angles in mathematics because they directly relate to the radius of a circle.

For equations like \( \sin \theta = \frac{\sqrt{2}}{2} \), solutions are often expressed in radians. Here, the basic solutions \( \theta = \frac{\pi}{4} \) and \( \theta = \frac{3\pi}{4} \) are each \( 45^\circ \) and \( 135^\circ \) when converted to degrees. Understanding this conversion is key in many branches of science where calculations are naturally performed in radians. For instance, when you solve \( \theta = \theta_0 + 2k\pi \), it represents an extension of the solution in terms of full circle rotations, expressed in radians.