Problem 28 - Precalculus with Limits

Question

In Exercises 19-28, find the exact solutions of the equation in the interval \( [0, 2\pi) \). \( \tan 2x - 2 \cos x = 0 \)

Step-by-Step Solution

Answer

The exact solutions of the equation are \(x = 0\) and \(x = \pi\)

Step 1: Converting \(\tan 2x\) to Sine and Cosine

The given equation is \( \tan 2x - 2 \cos x = 0 \). We can convert \(\tan 2x\) into sine and cosine functions using the double-angle identity: \( \tan 2x = \frac{2 \sin x}{1 - \sin^2 x} \) So the equation becomes: \( \frac{2 \sin x}{1 - \sin^2 x} - 2 \cos x = 0 \)

Step 2: Simplifying the Equation

Remember \(\cos^2 x = 1 - \sin^2 x\), substitute it into the equation and bringing all the terms to one side of the equation will result to: \( 2 \sin x - 2( 1 - \sin^2 x) = 0 \) Simplifying this gives: \( 2 \sin^3 x -2\sin x +2 = 0 \) Or equivalent to: \( \sin^3 x -\sin x +1 = 0 \)

Step 3: Solving the Equation for x

Now one can solve the equation \( \sin^3 x -\sin x +1 = 0 \) graphically or by trying some known values of x for which certain values of sin(x) are known (0, \(\pi/2\), \(\pi\), \(3\pi/2\)). Upon trying these values, one will observe that the equation holds true only for x = 0 or \(x = \pi\).

Key Concepts

Double-Angle Identities

Double-angle identities in trigonometry are formulas that allow us to express trigonometric functions of double angles, such as \(2x\), in terms of single angles like \(x\). These identities are crucial for simplifying and solving trigonometric equations that involve angles with multiples. The basic double-angle identities for sine, cosine, and tangent are:

• \( \sin(2x) = 2\sin(x)\cos(x) \)
• \( \cos(2x) = \cos^2(x) - \sin^2(x) \) which can also be written as \( 1 - 2\sin^2(x) \) or \( 2\cos^2(x) - 1 \)
• \( \tan(2x) = \frac{2\tan(x)}{1 - \tan^2(x)} \)

In the exercise, the function \(\tan(2x)\) is involved, and to solve the equation, we use the double-angle identity for the tangent to convert \(\tan(2x)\) into expressions involving \(\sin(x)\) and \(\cos(x)\). This maneuver simplifies the equation, moving us closer to finding the solutions for the variable \(x\). Applying these identities effectively can often be the key to solving complex trigonometric problems.

Solving Trigonometric Equations

Solving trigonometric equations involves finding the values of the angles that satisfy the given trigonometric expression. These equations can often be made simpler by employing trigonometric identities, as was done in the original exercise using a double-angle identity.

When presented with an equation like \( \tan 2x - 2 \cos x = 0 \), one effective strategy is to express all trigonometric functions in terms of a single function, such as \(\sin(x)\) or \(\cos(x)\), which is what we observe in the step-by-step solution. By transforming the equation accordingly, you reduce the complexity and isolate the variable \(x\).

Once the terms are organized and simplified, you may end up with a polynomial equation. In these cases, factoring, using known sine or cosine values, or graphing the function are all viable methods to solve for \(x\). In the textbook solution, known values of \(x\) that correspond to simple sine values are tested, leading to the discovery of the solutions within the specified interval \([0, 2\pi)\).

Trigonometric Functions

Trigonometric functions are mathematical functions of an angle, and they are foundational in the study of triangles and modeling periodic phenomena. The primary trigonometric functions are sine (\(\sin\)), cosine (\(\cos\)), and tangent (\(\tan\)), each of which have specific meanings and interpretations in terms of a right-angle triangle or a unit circle.

• The \(\sin(x)\) function represents the ratio of the length of the opposite side to the hypotenuse in a right-angled triangle.
• The \(\cos(x)\) function represents the ratio of the length of the adjacent side to the hypotenuse.
• The \(\tan(x)\) function represents the ratio of the length of the opposite side to the length of the adjacent side.

In the context of a unit circle, \(\sin(x)\) and \(\cos(x)\) correspond to the y-coordinate and x-coordinate of a point on the circle, respectively. These functions have specific values at standard angles, and they have characteristic graphs known as sinusoidal waves. Trigonometric functions are periodical, meaning they repeat values over regular intervals, which is a key aspect to consider when solving trigonometric equations.