Problem 19 - Algebra 2

Question

Find the exact values of the six trigonometric functions of \(\theta\) if the terminal side of \(\theta\) in standard position contains the given point. \((-1,0)\)

Step-by-Step Solution

Answer

Sine: 0, Cosine: -1, Tangent: 0, Cosecant: Undefined, Secant: -1, Cotangent: Undefined.

Step 1: Identify the Point and its Position

The given point is (-1, 0), which is located on the x-axis. Since it is specifically on the negative x-axis, the angle \(\theta\) is at \(180^\circ\) or \(\pi\) radians.

Step 2: Determine the Circle and Reference Radius

Since the point lies on the x-axis, the circle centered at the origin (0,0) with radius 1 (unit circle) will have the point (-1,0) on its circumference. The radius is 1 (since \(\sqrt{x^2 + y^2} = \sqrt{(-1)^2 + 0^2} = 1\)).

Step 3: Calculate Sine and Cosine Values

The cosine of an angle on the unit circle is the x-coordinate of the point, and the sine is the y-coordinate. For \((-1, 0)\), \(\cos(\theta) = -1\) and \(\sin(\theta) = 0\).

Step 4: Calculate the Tangent Value

Tangent is defined as the sine divided by the cosine: \(\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} = \frac{0}{-1} = 0\).

Step 5: Calculate the Reciprocal Trigonometric Functions

Cosecant, secant, and cotangent are the reciprocals of sine, cosine, and tangent respectively. Since \(\sin(\theta) = 0\), cosecant \(\csc(\theta) = \frac{1}{0}\) is undefined. \(\sec(\theta) = \frac{1}{\cos(\theta)} = \frac{1}{-1} = -1\). \(\cot(\theta) = \frac{1}{\tan(\theta)} = \frac{1}{0}\) is also undefined.

Key Concepts

Understanding the Unit Circle

The unit circle is a crucial concept in trigonometry because it visually represents trigonometric functions using a circle with a radius of 1 centered at the origin of a coordinate plane. Whenever you hear 'unit circle,' think of a circle with these key properties:

• A center at the point (0,0).
• Radius equals 1.
• Points on the circle have coordinates (\( \cos(\theta) \), \( \sin(\theta) \)), representing cosine and sine values respectively.

The unit circle helps in easily finding the sine, cosine, and tangent values of angles as it makes these relationships direct and visual. For any given angle \(\theta\), its corresponding point on the unit circle reveals both the cosine and sine values through its x- and y-coordinates. This is particularly helpful for angles like 0, 90, 180, and 270 degrees, where the points are precisely on the x- or y-axis at unit distance.

Angles in Standard Position

Angles in standard position are a foundational concept in trigonometry that helps in uniformly determining an angle's measure based on its terminal side. An angle is said to be in standard position if:

• Its vertex is placed at the origin of the coordinate plane.
• Its initial side lies along the positive x-axis.

The terminal side is the arm of the angle that rotates, and its position allows you to measure the angle. Starting from the initial side, if the terminal side lands in the same position as the point (-1,0) in the original exercise, this angle measures 180 degrees or \(\pi\) radians. Understanding this concept simplifies identifying and comparing angles as they all start from the same baseline, making angle calculation more straightforward.

Reciprocal Trigonometric Functions

Reciprocal trigonometric functions are remarkably useful in various trigonometric calculations. They extend beyond the basic sine, cosine, and tangent functions to include their reciprocals:

• Cosecant \(\csc(\theta)\): Reciprocal of sine (\(\sin(\theta)\)), defined as \(\csc(\theta) = \frac{1}{\sin(\theta)}\).
• Secant \(\sec(\theta)\): Reciprocal of cosine (\(\cos(\theta)\)), given by \(\sec(\theta) = \frac{1}{\cos(\theta)}\).
• Cotangent \(\cot(\theta)\): Reciprocal of tangent (\(\tan(\theta)\)), expressed as \(\cot(\theta) = \frac{1}{\tan(\theta)}\).

While they provide alternate means of representing these trigonometric values, it's crucial to note cases where certain functions become undefined. For instance, \(\csc(\theta)\) becomes undefined wherever \(\sin(\theta) = 0\), as seen with our angle \(\theta = 180^\circ\), affecting both \(\csc\) and \(\cot\). Mastery of these reciprocal functions enhances your ability to solve various trigonometric problems, especially where division or inversion of typical sine, cosine, and tangent functions is required.