Problem 28 - Technical Mathematics with Calculus

Question

Find the zeros in one cycle of the given functions, and the value of \(y\) at the given value of \(x\).$$y=4 \sin (x+2) \quad x=1 \text { radian }$$

Step-by-Step Solution

Answer

The zeros of one cycle of the function \(y = 4 \sin (x + 2)\) are at \(x = -2\), \(x = -2 + \pi\), and \(x = -2 - \pi\), but within the range [0, 2\pi], there are no zeros. The value of \(y\) at \(x = 1\) radian is approximately \(y = 4 \sin(3)\).

Step 1: Determine the zeros of the function

To find the zeros within one cycle of the function, set the equation equal to zero and solve for x: \(0 = 4 \sin (x + 2)\). Divide both sides by 4 to isolate the sine function: \(0 = \sin (x + 2)\). The zeros of the sine function occur when the angle is a multiple of \(\pi\), so set \(x + 2 = n\pi\) for \(n\) being an integer. To find zeros within one cycle, we consider \(n = 0\) and \(n = \pm1\), which give us \(x = -2\), \(x = -2 + \pi\), and \(x = -2 - \pi\). However, since we are looking within one cycle ([0, 2\pi]), we only consider the solutions that fall within that range.

Step 2: Find the value of y at x = 1 radian

Plug in \(x = 1\) radian into the given function: \(y = 4 \sin (1 + 2)\). Evaluate the sine function at \(3\) radians to get the value of \(y\). Since \(\sin(3)\) is not a standard angle on the unit circle, you may use a calculator to approximate its value.

Key Concepts

Trigonometric Zeros

Understanding trigonometric zeros is crucial when solving trigonometric functions. A 'zero' in trigonometry refers to the angle at which the trigonometric function yields a value of zero. For the sine function, zeros occur at integer multiples of \(\pi\), which corresponds to the angles where the sine wave intersects the horizontal axis on a graph.

When tackling a problem like \(y=4 \sin (x+2)\), you first set \(y\) to zero to establish the points where the function intersects the x-axis: \(0 = 4 \sin (x + 2)\). Simplifying this, we find zeros when \(\sin (x + 2) = 0\), and we identify these angles using the formula \(x + 2 = n\pi\), where \(n\) is any integer. However, because a 'cycle' of a sine function spans \(0\) to \(2\pi\) radians, one must consider only the solutions within this range to find the zeros in one cycle.

Sine Function

The sine function, one of the fundamental trigonometric functions, represents the vertical component of a point rotating on a unit circle. It's given by the equation \(y = \sin(x)\), where \(y\) varies between -1 and 1, creating a wave-like pattern known as a sine wave when graphed.

In problems where we have a modified sine function, such as \(y=4 \sin (x+2)\), knowing this pattern helps understand how the function behaves. The coefficient '4' indicates that the amplitude (height) of the sine waves has been multiplied by four. The \((x+2)\) shows a horizontal shift to the left by 2, altering where the function crosses the x-axis and achieves its maximum and minimum values.

Unit Circle

The unit circle is an essential concept in trigonometry, defined as a circle with a radius of 1 unit centered at the origin of a coordinate plane. It's a powerful tool for understanding trigonometric functions. The circle's circumference represents angles in a coordinate system where the horizontal axis is the cosine of an angle, and the vertical axis is the sine.

When you plot an angle on the unit circle, the x-coordinate of the point where the terminal side intersects the circle corresponds to the cosine of that angle, and the y-coordinate corresponds to the sine. This is how trigonometric functions arise from rotations and are related to circular motion. Hence, for angles not commonly found on the unit circle, such as \(3\) radians in our example, a calculator may be needed to determine the sine value.

Radial Measurement

Angles in trigonometry can be measured in degrees or radians—the latter being the standard unit of angular measurements in mathematics, denoted as 'rad'. One complete revolution around a circle is \(2\pi\) radians or 360 degrees. Thus, radians relate the arc length of a circle's sector to the radius of the circle, with 1 radian being the angle where the arc length is equal to the radius.

The use of radians is essential in understanding the periodicity and zeros of trigonometric functions. For example, when looking for zeros of the sine function, we use multiples of \(\pi\) radians to find standard positions where the sine value is zero. In the example problem, expressing angles in radians enables us to use \(x=1\) radian to easily compute the value of \(y\) using the sine function.