Problem 2 - Algebra and Trigonometry

Question

The basic equation \(\sin x=2\) has__________(no/one/infinitely many) solutions, whereas the basic equation \(\sin x=0.3\) has_________(no/one/infinitely many) solutions.

Step-by-Step Solution

Answer

The equation \(\sin x = 2\) has no solutions, while \(\sin x = 0.3\) has infinitely many solutions.

Step 1: Understand the Sine Function Range

The basic range of the sine function, \( \sin x \), is from -1 to 1. This means that for any real number \( x \), \( \sin x \) can only take values within the interval \([-1, 1]\).

Step 2: Analyze \(\sin x = 2\)

Since \( 2 \) is outside the range \([-1, 1]\), the equation \( \sin x = 2 \) has no real solutions because the sine of any angle cannot be 2.

Step 3: Analyze \(\sin x = 0.3\)

The value \( 0.3 \) is within the range \([-1, 1]\), so the equation \( \sin x = 0.3 \) has solutions. In fact, for any such value, there are infinitely many solutions due to the periodic nature of the sine function.

Key Concepts

Sine Function

The sine function, often denoted as \( \sin x \), is one of the fundamental trigonometric functions.- It is defined in the context of a right-angled triangle as the ratio of the length of the side opposite the angle to the length of the hypotenuse.- In the unit circle, \( \sin x \) represents the y-coordinate of a point on the circle corresponding to an angle \( x \) measured from the positive x-axis.
- The sine function is continuous and smooth, meaning it does not have any breaks or sharp turns.

Because the sine function is so integral to trigonometry and calculus, it is vital to understand its properties and behavior during computations and problem-solving processes.

Range of Sine Function

The range of a function is the set of possible output values it can produce. For the sine function, this range is specifically confined to the interval \([-1, 1]\).

- This means no matter what value \( x \) takes, the output of \( \sin x \) will always be between -1 and 1, inclusive.- This limitation arises because when the sine function is considered over a unit circle, it can only reach the maximum and minimum values at the topmost and bottommost points of the circle, corresponding to \(+1\) and \(-1\) respectively.
- Therefore, equations involving \( \sin x \) equal to numbers outside this range will have no real solutions, as such numbers are unattainable by the function.

Understanding the range is crucial for solving trigonometric equations, as it helps determine the possibility of solutions within the context of real numbers.

Periodic Functions

Periodic functions are functions that repeat their values in regular intervals or periods. The sine function is a prime example of a periodic function.- The period of the sine function is \( 2\pi \), meaning every \( 2\pi \) units along the x-axis, the function reverts back to its initial cycle of values. - This characteristic gives rise to the concept of infinitely many solutions in equations like \( \sin x = 0.3 \), since every distant interval of \( 2\pi \) will produce the same value of 0.3 again.
- Periodicity is a powerful property, as it allows the extension of trigonometric solutions beyond a narrow range, providing a broader spectrum of answers based on rotational symmetries observed in circles.

Grasping the periodic nature of functions like sine is essential for solving equations and understanding wave-like phenomena widely apparent in physics and engineering.