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Chapter 6: Problem 30

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

Short Answer

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

Step by step solution

01

Isolate the Trigonometric Function

First start by isolating the \(\sin x\) in the equation. To do that subtract 1 from both sides to get: \(\sqrt{2} \sin x = -1\)
02

Divide by \(\sqrt{2}\)

Now divide both sides by \(\sqrt{2}\) to get \(\sin x\): \(\sin x = -\frac{1}{\sqrt{2}}\), which simplifies to \(\sin x = -\frac{\sqrt{2}}{2}\).
03

Applying Inverse Trigonometric Function

Apply the arcsine (inverse sine) function to both sides. This gives, \(x = \sin^{-1}\left(-\frac{\sqrt{2}}{2}\right)\).
04

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 \(x = 2\pi - \frac{\pi}{4}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

Write the trigonometric expression as an algebraic expression. $$\sin (2 \arccos x)$$

Perform the addition or subtraction and simplify. $$\frac{2 x}{x^{2}-4}+\frac{5}{x+4}$$

Rewrite the expression in terms of \(\sin \theta\) and \(\cos \theta\) $$\frac{\csc \theta(1+\cot \theta)}{\tan \theta+\cot \theta}$$

The graph of a function \(f\) is shown over the 122, the graph of a function \(f\) is shown over the interval \([\mathbf{0}, \mathbf{2} \pi] .\) (a) Find the \(x\) -intercepts of the graph of \(f\) algebraically. Verify your solutions by using the zero or root feature of a graphing utility. (b) The \(x\) -coordinates of the extrema of \(f\) are solutions of the trigonometric equation. (Calculus is required to find the trigonometric equation.) Find the solutions of the equation algebraically. Verify these solutions using the maximum and minimum features of the graphing utility. Function: \(f(x)=\cos 2 x+\sin x\) Trigonometric Equation: \(-2 \sin 2 x+\cos x=0\)

Use the half-angle formulas to determine the exact values of the sine, cosine, and tangent of the angle. $$\frac{5 \pi}{8}$$
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