Problem 69 - Precalculus

Question

Determine whether each statement is true or false. \(\sin \theta=1\) when \(\theta=\frac{(2 n+1) \pi}{2}, n\) an integer.

Step-by-Step Solution

Answer

True, \(\sin \theta = 1\) for \(\theta = \frac{(2n+1)\pi}{2}\) when \(n\) is even.

Step 1: Understand the Sine Function

The sine function, \(\sin \theta\), represents the y-coordinate on the unit circle corresponding to the angle \(\theta\). The sine value reaches its maximum of 1 when \(\theta = \frac{\pi}{2} + 2k\pi\), where \(k\) is an integer.

Step 2: Analyze the Given Angle Formula

The angle \(\theta = \frac{(2n+1)\pi}{2}\) describes positions at odd multiples of \(\frac{\pi}{2}\). For these positions, when \(\theta = \frac{\pi}{2}, \frac{3\pi}{2}, \frac{5\pi}{2}, \ldots\), the sine value is alternates between 1 and -1.

Step 3: Determine When Sine Equals One

Review: Sine equals 1 specifically at \(\theta = \frac{\pi}{2} + 2k\pi\) for integer \(k\). In the formula \(\theta = \frac{(2n+1)\pi}{2}\), when \(n = 0, 2, 4, \ldots\) (even integers), \(\theta\) corresponds to positions like \(\frac{\pi}{2}, \frac{5\pi}{2}\), etc., where sine is 1.

Step 4: Decide the Truth of the Given Statement

The statement claims \(\sin \theta = 1\) for the formula \(\theta = \frac{(2n+1)\pi}{2}\). This is true when \(n\) is even, as explained. Since the claim doesn't specify the need for \(n\) to be even, the statement is overall true since \(n\) can encompass even integers.

Key Concepts

Unit Circle

The unit circle is a vital concept in trigonometry that helps us understand the behavior of trigonometric functions. It's a circle with a radius of 1, centered at the origin of the coordinate plane. The circumference of the unit circle represents all possible angles from 0 to 360 degrees, or from 0 to \(2\pi\) radians.

• The x-coordinate of a point on this circle represents the cosine of the angle.
• The y-coordinate represents the sine of the angle.

When we talk about the sine function using the unit circle, we refer to the vertical position of a point on this circle. As the angle \(\theta\) increases, the sine value traces the y-coordinate of the circle, reaching its maximum value of 1 when the point is at the top of the circle (directly above the origin). This happens at an angle of \(\frac{\pi}{2}\) radians, as visualized when the circle is rotated in positive (counter-clockwise) direction.

Angle Formula

The angle formula is crucial for understanding trigonometric functions in terms of their periodicity and repetition over the circle. The formula \(\theta = \frac{(2n+1)\pi}{2}\) encompasses angles that are odd multiples of \(\frac{\pi}{2}\). This pattern involves a sequence that includes angles such as:

• \(\theta = \frac{\pi}{2}, \frac{3\pi}{2}, \frac{5\pi}{2}, \ldots\)

For these angles, the sine function value toggles between 1 and -1.
This happens because each odd multiple results in alternating positions on the unit circle. For example:- At \(\frac{\pi}{2}\), the sine value is 1.- At \(\frac{3\pi}{2}\), the sine value becomes -1.
In essence, this formula captures the characteristic wiggle-pattern seen in the sine function as it cycles through the unit circle.

Integer Multiples

Understanding integer multiples in trigonometry, especially regarding angles, helps explain how sine and other functions repeat predictably on the unit circle. The equation \(\theta = \frac{(2n+1)\pi}{2}\) indicates that the angle \(\theta\) will vary based on the integer value \(n\). Here, \(n\) consists of all integers, including zero and negative numbers.

• If \(n\) is an even integer, like 0 or 2, this places \(\theta\) at positive peaks of the sine curve (where sine equals 1).
• If \(n\) is an odd integer, it places \(\theta\) at negative peaks (sine equals -1).

Thus, the statement \(\sin \theta = 1\) is correct for even \(n\), as the angle aligns with positions at the top of the unit circle. The formula \( \theta = \frac{(2n+1)\pi}{2} \) encompasses this repetition by illustrating the relationship between \(n\) and the sine wave's positive peaks. This helps solidify the sine function's periodic nature.