Problem 18 - Algebra and Trigonometry

Question

Find all solutions of each equation. $$ \sin x=0 $$

Step-by-Step Solution

Answer

The solutions for the equation \(\sin x = 0\) are \(x = n\pi\), where \(n\) represents all integer values.

Step 1: Identify the Function

The given function is a trigonometric function \(\sin x = 0\).

Step 2: Find Angles Where Sine Function Equals Zero

The sine function equals zero at two points within one cycle (0 to \(2\pi\) in radians or 0 to \(360º\) in degrees): \(0\) and \(\pi\) in radians or \(0º\) and \(180º\) in degrees.

Step 3: Consider All Cycles of Sine Function

Considering all cycles of the sine function, it crosses the x-axis at every multiple of \(\pi\) or \(180º\). The solutions to the equation are therefore \(x = n\pi\), where \(n\) is an integer (this can be positive or negative).

Key Concepts

Understanding the Sine Function

The sine function, denoted by \( \sin \theta \), is one of the fundamental trigonometric functions. It is most commonly used in the study of periodic phenomena, such as sound and light waves, because it describes smooth repetitive oscillations. The output of the sine function for an angle \( \theta \) represents the ratio of the length of the opposite side to the hypotenuse in a right-angled triangle. Key characteristics of the sine function include:

• The sine of an angle ranges between -1 and 1.
• It is periodic with a period of \( 2\pi \) radians (or 360 degrees), which means it repeats its values every \( 2\pi \) units.
• The graph of \( \sin x \) is a smooth wave that crosses the x-axis at multiples of \( \pi \).

Understanding these basic properties helps in solving equations involving \( \sin x \) and predicting where it might equal zero or any other value.

Solutions of Equations Involving Sine Function

Solving trigonometric equations, like \( \sin x = 0 \), involves finding all possible angles \( x \) that satisfy the equation. For \( \sin x = 0 \), the task is to figure out at which points the sine function equals zero.In one complete cycle of \( \sin x \) (from 0 to \( 2\pi \) radians), the function crosses the x-axis at:

• 0 radians (0 degrees)
• \( \pi \) radians (180 degrees)

These points are then repeated every cycle. This leads to the general solution:
\[ x = n\pi \]Where \( n \) is any integer. This equation expresses the fact that \( \sin x \) equals zero at every full rotation of \( \pi \) radians from the origin, both forwards and backwards along the number line.

Using Trigonometric Identities to Solve Equations

Trigonometric identities are handy tools for solving and simplifying trig equations. They are equations involving trigonometric functions that are true for all values of the occurring variables where both sides are defined. Examples include the Pythagorean identity \( \sin^2 x + \cos^2 x = 1 \) and the angle sum and difference identities.For the equation \( \sin x = 0 \), the identity or transformation isn't directly needed for finding the solutions, but understanding identities like \( \sin(\pi - x) = \sin x \) can help identify symmetries and various angles with equivalent sine values. Trig identities also become crucial when dealing with more complex trigonometric equations.

Understanding Radians and Degrees

Radians and degrees are two units for measuring angles. They are essential in trigonometry, each having its unique benefits.

• Degrees: More intuitive for novice learners and visualizing angles, as it breaks a full circle into 360 parts. Useful in navigation and circular motions known from everyday life.
• Radians: More mathematically versatile, describing angles in terms of the radius of a circle. A full circle is \( 2\pi \) radians. Radians often simplify formulas, particularly in calculus and physics.

To solve equations like \( \sin x = 0 \), we often toggle between these units as needed. The conversion is straightforward: \[ 180^\circ = \pi \text{ radians} \]
Understanding this conversion helps in checking solutions across different contexts and applications, ensuring flexibility in solving trigonometric problems.