Problem 70 - Algebra and Trigonometry

Question

Use an identity to solve each equation on the interval \([0,2 \pi)\) $$ \sin 2 x=\sin x $$

Step-by-Step Solution

Answer

The solution for the equation \(\sin 2x = \sin x\) in the interval \([0, 2\pi]\) are \(x = 0, \pi, \frac{\pi}{3}, \frac{5\pi}{3}\).

Step 1: Apply Trigonometric Identity

The equation is given by \(\sin 2x = \sin x\). We can apply trigonometric identity for \(\sin 2x\) which is \(2 \sin x \cos x\). Thus the equation becomes \(2 \sin x \cos x = \sin x\).

Step 2: Simplify the equation

To simplify, the \(\sin x\) should be factorized out of the equation: \(2 \sin x \cos x - \sin x = 0\Rightarrow \sin x (2 \cos x - 1) = 0\) Now, for a product to be equal to zero, either of the factors has to be zero: Thus, \( \sin x = 0\) or \(2 \cos x - 1 = 0\)

Step 3: Solve for \(x\)

For \(\sin x = 0\), within the interval \([0, 2\pi]\) , \(x = 0, \pi\). For \( 2\cos x - 1 = 0\), solving gives \(\cos x = \frac{1}{2}\). Within the interval \([0, 2\pi]\), the values that satisfy this equation are \(x= \frac{\pi}{3}, \frac{5\pi}{3}\).

Key Concepts

Trigonometric Identities

Understanding trigonometric identities is crucial to solving equations involving trigonometric functions. A trigonometric identity is an equality that holds true for all values of the variables involved. These identities are often used to simplify trigonometric expressions or convert one type of function into another.

For example, the identity \(\sin (2x) = 2\sin x \cos x\) is known as the double angle identity for sine. This identity is very useful for solving trigonometric equations because it allows us to express the sine of a double angle in terms of the sine and cosine of the original angle. In our exercise, the identity \(\sin (2x) = 2\sin x \cos x\) transforms the equation \(\sin 2x = \sin x\) into a form that is easier to solve.

Knowing the basic identities and when to apply them can immensely simplify complicated trigonometric equations. Identities like the Pythagorean identity (\(\sin^2 x + \cos^2 x = 1\)), the angle sum and difference identities, and the co-function identities are all fundamental tools in a mathematician's toolkit.

Sine Function

The sine function, represented as \(\sin x\), is one of the primary trigonometric functions. It relates the angle of a right triangle to the ratio of the length of the opposite side to the length of the hypotenuse. The sine function is periodic, with a period of \(2\pi\) radians (or 360 degrees), meaning it repeats its values in a regular pattern every \(2\pi\) radians.

In the context of our exercise, we encounter the equation \(\sin x = 0\). The zeros of the sine function occur at integer multiples of \(\pi\), where the angle corresponds to a position on the x-axis when considering the unit circle. For the interval \( [0, 2\pi] \), the values that satisfy \(\sin x = 0\) are \(x = 0\) and \(x = \pi\). It's essential to understand the behavior of the sine function to predict its values for specific angles and solve equations effectively.

Cosine Function

Parallel to the sine function, the cosine function, denoted as \(\cos x\), plays a significant role in trigonometry. The cosine function gives the ratio of the length of the adjacent side to the hypotenuse in a right-angled triangle for a given angle. Like the sine function, the cosine function is also periodic, with a period of \(2\pi\) radians.

In solving the equation \(2\cos x - 1 = 0\) from our exercise, we're primarily concerned with finding the angles where the cosine value is \(\frac{1}{2}\). These angles are found by considering the unit circle and recognizing that cosine values are positive in the first and fourth quadrants. Specifically, for the range \( [0, 2\pi] \), the angles at which \(\cos x = \frac{1}{2}\) are \(x = \frac{\pi}{3}\) and \(x = \frac{5\pi}{3}\). Recognizing these patterns in the cosine function's graph can help to quickly identify solutions to trigonometric equations involving cosine.