Problem 20 - Contemporary Precalculus

Question

Use your knowledge of special values to find the exact solutions of the equation. $$\sin x=0$$

Step-by-Step Solution

Answer

Answer: The sine function equals zero at all angles in the form of x = nπ, where n is an integer (positive, negative, or zero).

Step 1: Identify the sine function's values at special angles

We need to find the angles at which the sine function is equal to zero. For this, let's consider some special angles: \(0^\circ\), \(90^\circ\), \(180^\circ\), and \(270^\circ\) (or in radians \(0\), \(\frac{\pi}{2}\), \(\pi\), and \(\frac{3\pi}{2}\)). We can find that: $$\sin 0^\circ = \sin 180^\circ = 0$$ $$\sin 90^\circ \neq 0 \text{ and } \sin 270^\circ \neq 0$$ Using these special angles, we can see that the sine function equals zero at multiples of \(180^\circ\) (without including \(90^\circ\) and \(270^\circ\) multiples).

Step 2: Generalize the solution for all angles

Now that we know the sine function equals zero at multiples of \(180^\circ\) or in radians, multiples of \(\pi\), we can generalize this to include all angles where \(\sin x = 0\). The general solution for the given equation is: $$x = n\pi$$ where \(n\) is an integer (positive, negative, or zero).

Key Concepts

Sine Function

The sine function is a fundamental aspect of trigonometry that relates to the ratio of the opposite side to the hypotenuse in a right-angled triangle. It is also represented as a wave on a coordinate plane, oscillating between -1 and 1 on the y-axis. The function is periodic with a period of \(2\pi\) radians or 360 degrees, meaning it repeats itself after each complete cycle.

Key characteristics include:

• The sine of 0 degrees or 0 radians is 0.
• The sine function reaches its maximum of 1 at 90 degrees or \(\frac{\pi}{2}\) radians.
• Its minimum of -1 occurs at 270 degrees or \(\frac{3\pi}{2}\) radians.

Understanding the sine function is crucial in solving trigonometric equations, especially knowing where it hits specific values like zero.

Special Angles

Special angles are specific angles in trigonometry that have known sine, cosine, and tangent values. These angles include 0, 30, 45, 60, and 90 degrees or their radian equivalents. For sine, these angles provide well-known results that simplify solving equations.

For the equation \(\sin x = 0\), the most critical special angles are 0 degrees and 180 degrees because:

• \(\sin 0^\circ = 0\)
• \(\sin 180^\circ = 0\)

These values repeat every 180 degrees or \(\pi\) radians, which guides us in forming general solutions. Recognizing these angles saves time and simplifies finding solutions to trigonometric equations.

Radians

Radians offer another way to measure angles, often more natural in mathematics than degrees. One radian is the angle formed by taking the radius of a circle and wrapping it along the circle's edge. Consequently, a complete rotation (360 degrees) equals \(2\pi\) radians.

Converting between radians and degrees is vital:

• To convert degrees to radians, multiply by \(\frac{\pi}{180}\).
• To convert radians to degrees, multiply by \(\frac{180}{\pi}\).

In the context of the sine equation \(\sin x = 0\), recognizing angles in radians ensures we utilize special angles appropriately, such as 0, \(\pi\), and \(2\pi\), where the sine function is zero.

General Solutions

When solving trigonometric equations, finding general solutions broadens the scope to include all possible solutions. Trigonometric functions like sine are periodic, meaning their values repeat in a regular cycle.

For \(\sin x = 0\), the sine function equals zero at every multiple of \(\pi\) radians:

• This can be represented as \(x = n\pi\).
• Here, \(n\) is an integer, allowing for positive, negative, or zero values.

Understanding how to derive the general solution ensures that no potential solutions are overlooked, giving a complete answer to the problem. This concept is essential for covering not only specific cases but the entire set of solutions.