Problem 12 - Algebra 2

Question

Find the exact value of each expression. If the expression is undefined, write undefined. $$ \sec 180^{\circ} $$

Step-by-Step Solution

Answer

\(-1\)

Step 1: Identify the Trigonometric Function

The secant function is the reciprocal of the cosine function. So sec(x) is equal to 1/cos(x). Hence, we need to find the value of cos(180 degrees).

Step 2: Find the Value of Cosine

The value of cos(180 degrees) is -1 as per the unit circle and basic principles of Trigonometry.

Step 3: Find the Reciprocal

Finally, we know that sec(180 degrees) = 1/cos(180 degrees). So, substituting the value we have sec(180 degrees) = 1/(-1), we conclude that the result is -1.

Key Concepts

Secant Function

The secant function is one of the six primary trigonometric functions used in trigonometry. It is symbolized as "sec" and is defined mathematically as the reciprocal of the cosine function. This means that for an angle \( x \), the secant function is expressed as:

• \( \sec(x) = \frac{1}{\cos(x)} \)

Knowing the secant function is fundamental because it helps in solving various trigonometric equations, especially those involving the cosine function. The secant function is undefined if the cosine of the angle is zero, as division by zero is impossible.
For example, this occurs at angles like 90° and 270°, where the cosine is zero. The secant function is crucial while working with angles in both degrees and radians, and it plays an essential role in trigonometric identities and equations.

Unit Circle

The unit circle is a powerful tool in trigonometry that allows us to understand the relationships between different trigonometric functions and angles. It is a circle with a radius of one, centered at the origin of a coordinate plane. The unit circle helps us find the sine, cosine, and their reciprocals for various angles.Important aspects of the unit circle include:

• The angle is measured from the positive x-axis, moving counter-clockwise.
• The coordinates of a point on the unit circle can be written as \((\cos(\theta), \sin(\theta))\).
• The unit circle can be used to determine trigonometric values for commonly known angles, such as 0°, 30°, 45°, 60°, and 90°, among others.

Knowing how to read and interpret the unit circle allows us to quickly find cosine and sine values for different angles. Moreover, it helps us visualize the periodic nature of the trigonometric functions and understand how these functions behave as angles increase or decrease.

Cosine Function

The cosine function is a fundamental trigonometric function that relates the angle of a right triangle to the length of the adjacent side over the hypotenuse. It is often denoted as "cos(x)", where \(x\) represents the angle. In the unit circle, the cosine of an angle corresponds to the x-coordinate of a point on the circle.Key points about the cosine function:

• It is periodic with a period of 360° (or \(2\pi\) radians).
• The range of the cosine function is between -1 and 1.
• Common cosine values include \( \cos(0°) = 1 \) and \( \cos(180°) = -1 \).

The cosine function is critical in determining both the secant function and other derived trigonometric functions. The unit circle shows us that for an angle of 180°, the cosine is -1, which is a vital part of finding the secant of 180°. This relationship illustrates the deep connection between various trigonometric functions, emphasizing the importance of mastering them for solving trigonometric problems.