Problem 30 - Algebra and Trigonometry Real Mathematics, Real People

Question

Solve the equation. $$\sqrt{2} \sin x+1=0$$

Step-by-Step Solution

Answer

The solutions to the equation are \(x = \pi + \frac{\pi}{4}\) and \(x = 2\pi - \frac{\pi}{4}\).

Step 1: Isolate the Trigonometric Function

First start by isolating the \(\sin x\) in the equation. To do that subtract 1 from both sides to get: \(\sqrt{2} \sin x = -1\)

Step 2: Divide by \(\sqrt{2}\)

Now divide both sides by \(\sqrt{2}\) to get \(\sin x\): \(\sin x = -\frac{1}{\sqrt{2}}\), which simplifies to \(\sin x = -\frac{\sqrt{2}}{2}\).

Step 3: Applying Inverse Trigonometric Function

Apply the arcsine (inverse sine) function to both sides. This gives, \(x = \sin^{-1}\left(-\frac{\sqrt{2}}{2}\right)\).

Step 4: Find the Value of x

To get the solution, remember that the sine is negative in the third and fourth quadrants. Thus, the solution in radians will be \(x = \pi + \frac{\pi}{4}\) and \(x = 2\pi - \frac{\pi}{4}\).

Key Concepts

Inverse Trigonometric Functions

Understanding inverse trigonometric functions is essential when solving trigonometric equations. These functions, like the arcsine (denoted as \( \sin^{-1} \)), are used to find the angle when the value of the trigonometric function is known. For instance, if we have \( \sin x = -\frac{\sqrt{2}}{2} \), applying the inverse sine function would allow us to solve for \(x\).

However, caution is necessary as trigonometric functions are periodic and can have multiple angle solutions. Since the \(\sin^{-1}\) function returns values in the range of \( -\frac{\pi}{2} \text{ to } \frac{\pi}{2} \), we must adjust for the specific quadrants where sine is negative to find all possible solutions.

Basic Trigonometric Identities

Trigonometric identities are the equalities involving trigonometric functions that are true for all values of the involved variables. Knowing these identities helps in transforming and simplifying trigonometric expressions and solving equations. A fundamental identity, for example, is \( \sin^2 x + \cos^2 x = 1 \), which expresses the inherent relationship between sine and cosine functions.

Besides this, there are other reciprocal, quotient, and Pythagorean identities. In solving equations, like \( \sqrt{2} \sin x + 1 = 0 \), we often use these identities to rearrange terms, simplify, or convert between trigonometric functions to find the solutions.

Radian Measure

The radian measure is a way of expressing angles, and it is the standard unit of angular measure used in many areas of mathematics. One radian is the angle subtended at the center of a circle by an arc that is equal in length to the radius of the circle. This concept becomes crucial when solving trigonometric equations, where solutions are often presented in radians.

When solving the given equation, after applying the inverse sine, we determine that the solutions are at \(\pi + \frac{\pi}{4}\) and \(2\pi - \frac{\pi}{4}\), which are both radian measures of the angles. Remembering the relationship between radians and degrees (\(2\pi\) radians equals \(360^\circ\)) can help contextualize these solutions on the unit circle.