Problem 16 - Precalculus Essentials

Question

Find all solutions of each equation. $$\sin x=-\frac{\sqrt{2}}{2}$$

Step-by-Step Solution

Answer

The solutions to the equation \(\sin x = -\sqrt{2}/2\) are \(x = k \cdot 2π + 3π/4 , x = k \cdot 2π + 7π/4\) where k is an integer.

Step 1: Identify sine values in unit circle

In a unit circle, sine of an angle is the y-coordinate. Sine is negative in the 3rd and 4th quadrants. When the angle forms an isosceles right triangle in the unit circle, the sine value equals -sqrt(2)/2. This occurs at angles 3π/4 and 7π/4.

Step 2: Identify solution in principle domain

The angles 3π/4 and 7π/4 are already in the principle domain [0,2π). So these are the two solutions in the principle domain.

Step 3: List all solutions

The sine function is periodic with period 2π. This means that for any solution x, all solutions will be of the form \(x = k \cdot 2π + Solution\) , where k is an integer and Solution represents the solutions from the principle domain. For the problem at hand, when putting it all together, the entire set of solutions is \(x = k \cdot 2π + 3π/4 , x = k \cdot 2π + 7π/4\) where k is an integer.

Key Concepts

Sine Function

The sine function is one of the fundamental functions in trigonometry. It is particularly significant due to its relationship with circles and oscillatory behavior, like waves. In a right triangle, the sine of an angle is the ratio of the length of the opposite side to the hypotenuse.
This function can take any real number as input and produces an output that ranges between -1 and 1. Due to this range, sine values are repeated in a periodic cycle.

• Sine of 0 degrees (or 0 radians) is 0.
• Sine of 90 degrees (or π/2 radians) is 1.
• Sine of 180 degrees (or π radians) is 0.
• Sine of 270 degrees (or 3π/2 radians) is -1.

Understanding the key values at these primary angles is essential for solving equations that involve the sine function.

Unit Circle

The unit circle is a circle with a radius of 1, centered at the origin of the coordinate plane. This circle is a powerful tool for visualizing and determining the values of trigonometric functions. When a point travels around the unit circle, the y-coordinate of this point represents the sine of the angle.

• In the first quadrant, angles have positive sine values.
• In the second quadrant, sine values are also positive.
• However, in the third and fourth quadrants, sine values are negative.

This is important because, when solving an equation like \(\sin x = -\frac{\sqrt{2}}{2}\), you know immediately that the angle must be in the third or fourth quadrant. Understanding the layout of the unit circle helps streamline solving trigonometric equations.

Angle Solutions

When dealing with the sine function, angle solutions are the specific angles that satisfy the given equation. For the equation \(\sin x = -\frac{\sqrt{2}}{2}\), we identify angles in specific quadrants.
Since the sine is negative, we look at the third and fourth quadrants of the unit circle. Each angle is measured in radians, which are units for angles based on the radius of a circle. Common solutions for \(\sin \) equations often involve π/4 or changes thereof.

• In the third quadrant, \ 5π/4 \ is an angle solution.
• In the fourth quadrant, \ 7π/4 \ is an angle solution.

These angles are found by noting that the sine value \ -\frac{\sqrt{2}}{2} \ appears at 45 degrees past the x-axis in both the third and fourth quadrants.

Periodic Functions

Periodic functions are those that repeat their values in regular intervals or periods. The sine function has a period of \(2π\), meaning its values repeat every \(2π\) radians. This is why we can create a general solution for sine equations by adding multiples of \(2π\) to specific solutions.
Once you've identified one solution, like \(\frac{3π}{4}\) or \(\frac{7π}{4}\), you can find all other solutions by adding or subtracting multiples of \(2π\). This concept gives rise to equations such as \(x = k(2π) + \text{Solution}\) where \(k\) is any integer.
This periodic nature is a fundamental property that makes trigonometric functions reliably repetitive, beneficial for applications involving cycles, waves, and rotations. Understanding periodicity is crucial for predicting and calculating trigonometric values efficiently.