Problem 119 - Precalculus

Question

Find the \(y\) -intercept of \(y=-A \sin \left(B x+\frac{\pi}{6}\right)\)

Step-by-Step Solution

Answer

The y-intercept is \(-\frac{A}{2}\).

Step 1: Understanding the y-intercept

The y-intercept of any function is the point where the graph of the function intersects the y-axis. This occurs when the value of \(x\) is 0. For the given sine function, the y-intercept is found by setting \(x = 0\) in the equation \(y = -A \sin \left(Bx + \frac{\pi}{6}\right)\).

Step 2: Substitute x with 0

Substitute \(x = 0\) into the equation: \(y = -A \sin \left(B \times 0 + \frac{\pi}{6}\right)\). This simplifies to: \(y = -A \sin \left(\frac{\pi}{6}\right)\).

Step 3: Calculate \( \sin \left(\frac{\pi}{6}\right) \)

The angle \( \frac{\pi}{6} \) corresponds to 30 degrees. The sine of 30 degrees, \( \sin \left(\frac{\pi}{6}\right) \), is \( \frac{1}{2} \).

Step 4: Find the y-intercept value

Substitute \( \sin \left(\frac{\pi}{6}\right) = \frac{1}{2} \) into the equation: \(y = -A \times \frac{1}{2} = -\frac{A}{2}\). Thus, the y-intercept is \(-\frac{A}{2}\).

Key Concepts

y-Intercept

The concept of the y-intercept is central to understanding graphs of equations. The y-intercept is the point where a graph intersects the y-axis. If you imagine the graph as a line or curve crossing the vertical axis on a graph, that crossing point is the y-intercept. For any mathematical function, finding the y-intercept involves setting the value of the horizontal axis, or x, to zero.

In the equation given in this example, which is a sine function in the form of \(y = -A \sin \left(Bx + \frac{\pi}{6}\right)\), determining the y-intercept requires plugging zero into the equation for x. This reduces the function down to \(y = -A \sin \left(\frac{\pi}{6}\right)\), where the remaining task is to solve for this expression to find the y-intercept. This fundamental technique helps to identify how graphs behave at specific key points.

Sine Function

Sine functions are a type of trigonometric function known for their wave-like properties. The sine function, denoted as \sin(x)\, is a periodic function. This means it repeats its values in regular intervals or periods. It ranges between -1 and 1, reflecting its oscillating nature.

The equation \(y = -A \sin \left(Bx + \frac{\pi}{6}\right)\) describes a sine function. In this expression, A is the amplitude, which affects the height of the peaks and the depths of the troughs of the graph. The 'negative' sign in front of A indicates that the wave is inverted. B represents the frequency, dictating how many waves there are in a given length on the graph.

Understanding how each component of the sine function affects the graph is crucial. The angle addition, \(\frac{\pi}{6}\), shifts the wave horizontally, allowing for fine control over start positions and intercepts. Recognizing the role of each variable in the sine function helps predict and interpret its behavior in graphical representations.

Angle Conversion

Angle conversion is essential when working with trigonometric functions as angles can be measured in degrees or radians. These are simply two ways of expressing size of an angle. Radians are often used in mathematical problems because they connect angles to the arc length of a circle.

To convert from degrees to radians, use the formula: \text{Radians} = \text{Degrees} \times \frac{\pi}{180}\. The given problem mentions an angle of \(\frac{\pi}{6}\). This is the radian equivalent of 30 degrees, which is crucial because knowing and using the radian measure allows for seamless calculation in trigonometry problems.

This conversion becomes particularly important in the context of sine, cosine, and other trigonometric functions because they default to angles measured in radians. Understanding angle conversion ensures you can compute trigonometric values accurately, such as finding that \sin(30^\circ)\ equals \frac{1}{2}\ or \sin(\frac{\pi}{6}) = \frac{1}{2}\, maintaining consistency across different applications and problems.