Problem 28 - Algebra 2 and Trigonometry

Question

When \(\tan \theta=0,\) find a value of: \(\quad\) a. \(\sin \theta \quad\) b. \(\cos \theta \quad \mathbf{c} . \theta\)

Step-by-Step Solution

Answer

\( \sin \theta = 0 \), \( \cos \theta = 1 \), \( \theta = 0 \).

Step 1: Identifying \\( an heta = 0 \\)

When \( \tan \theta = 0 \), it implies that the tangent function value is zero. This occurs when \( \theta \) is at angles where \( heta = n \pi \), where \( n \) is an integer.

Step 2: Determine \( \sin \theta \)

At \( \theta = n \pi \), the sine function \( \sin \theta = 0 \). This is because at these angles, the y-coordinate on the unit circle is zero, resulting in sine being zero.

Step 3: Determine \( \cos \theta \)

At \( \theta = n \pi \), the cosine function \( \cos \theta = (-1)^n \). This means that when \( n \) is even, \( \cos \theta = 1 \), and when \( n \) is odd, \( \cos \theta = -1 \).

Step 4: Find a Value for \( \theta \)

Given that \( \tan \theta = 0 \) occurs at \( n \pi \), the simplest values are \( \theta = 0 \) or \( \theta = \pi \). For simplicity, we can choose \( \theta = 0 \).

Key Concepts

Tangent Function

The tangent function, often denoted as \( \tan \theta \), is a fundamental trigonometric function. It is primarily defined as the ratio of the sine and cosine functions. Mathematically, it can be expressed as:

• \( \tan \theta = \frac{\sin \theta}{\cos \theta} \)

The tangent function is periodic, with a period of \( \pi \) radians (or 180 degrees). This means that it repeats its values every \( \pi \) units along the angle θ.
In terms of graph behavior, \( \tan \theta \) takes values from negative infinity to positive infinity, and its graph exhibits vertical asymptotes where \( \cos \theta = 0 \). When \( \tan \theta = 0 \), it indicates that \( \sin \theta \) must be zero, as the sine numerator is zero when the tangent is zero.
This generally occurs where \( \theta = n\pi \), with \( n \) being an integer. \( \tan \theta \) equals zero at these points because the sine of these angles is zero, making the overall ratio zero.

Sine Function

The sine function, represented by \( \sin \theta \), describes the y-coordinate of a point on the unit circle that is a certain angle \( \theta \) from the positive x-axis. The sine function is periodic, with a cycle of \( 2\pi \) radians (or 360 degrees), which means its values repeat every \( 2\pi \) units.

• At angle \( \theta = n\pi \), the \( \sin \theta \) equals zero.
• This is because at these angles, the point on the unit circle lies directly on the x-axis.

Fundamentally, the sine function has a maximum value of +1 and a minimum value of -1, fluctuating between these tracks as the circle rotates through \( \theta \).
The sine function is critical in describing wave-like phenomena and is used extensively in physics and engineering to model oscillatory behavior.

Cosine Function

The cosine function, denoted \( \cos \theta \), refers to the x-coordinate of a point on the unit circle at a given angle \( \theta \). Like the sine function, the cosine function also has a period of \( 2\pi \) radians. This means it cycles through its values every \( 2\pi \) radians, returning to the initial values.
When considered at values like \( \theta = n\pi \):

• If \( n \) is even (for instance 0, 2, 4, etc.), \( \cos \theta = 1 \).
• If \( n \) is odd (such as 1, 3, 5, etc.), \( \cos \theta = -1 \).

The cosine function reaches its peak at 1 and the lowest at -1, providing a cosine wave pattern that is essential in trigonometry for analyzing cyclic behaviors and patterns in mathematical and real-world contexts.

Unit Circle

The unit circle is a critical concept in trigonometry and serves as a foundation for the definitions of the sine, cosine, and tangent functions. It is a circle with a radius of one centered at the origin (0,0) of a coordinate plane.
The circle is divided by angles \( \theta \), which typically range from 0 to \( 2\pi \) radians. As these angles traverse the circle, their sine and cosine determine the respective y-coordinates and x-coordinates of a point on the circle.
For any point on the unit circle:

• The equation \( x^2 + y^2 = 1 \) holds true.
• The x-value is equal to \( \cos \theta \) and the y-value is \( \sin \theta \).

The unit circle helps visualize the oscillation of the trigonometric functions and is a vital tool in exploring angular relationships and periodic phenomena. It also assists in understanding the behavior of functions as they relate to right triangles.

Angles

Angles are the fundamental measures in trigonometry and are formed by two rays that originate from a single vertex. Angles are most commonly measured in degrees, where a complete revolution equals 360 degrees, and radians, where a full circle corresponds to \( 2\pi \) radians.
Commonly used angles include 0, 30, 45, 60, 90, 180 degrees or their radian equivalents. These are essential for defining positions on the circle when working with trigonometric functions.
When considering trigonometric functions:

• A 0-degree angle (or 0 radians) suggests the positive x-axis, leading to \( \cos 0 = 1 \) and \( \sin 0 = 0 \).
• At angle \( \theta = n\pi \), where \( n \) is an integer, the corresponding tangent for these angles becomes zero.

Understanding angles is crucial for deciphering trigonometric problems, visualizing function graphs, and merging geometry and algebra in problem-solving scenarios.