Problem 27 - Algebra 2 and Trigonometry

Question

In \(3-44,\) find the exact value. $$ \sec 270^{\circ} $$

Step-by-Step Solution

Answer

The exact value of \( \sec 270^{\circ} \) is undefined.

Step 1: Understanding Secant Function

The secant function, denoted as \( \sec \theta \), is the reciprocal of the cosine function. This means that \( \sec \theta = \frac{1}{\cos \theta} \).

Step 2: Identify 270 Degrees in the Unit Circle

On the unit circle, 270 degrees corresponds to the point (0, -1), which is the negative y-axis.

Step 3: Calculate Cosine of 270 Degrees

The cosine of an angle is the x-coordinate of the corresponding point on the unit circle. For 270 degrees, \( \cos 270^{\circ} = 0 \).

Step 4: Find Secant of 270 Degrees

Since \( \sec \theta = \frac{1}{\cos \theta} \), substituting the value of cosine, we have \( \sec 270^{\circ} = \frac{1}{0} \), which is undefined.

Key Concepts

Secant Function

The secant function is a fundamental trigonometric function that relates directly to the cosine function. It is represented as \( \sec \theta \) and is defined as the reciprocal of the cosine function. This relationship can be expressed with the equation \( \sec \theta = \frac{1}{\cos \theta} \). Understanding this relationship is key to solving problems involving secant because it means wherever the cosine is zero, secant becomes undefined.

When working with trigonometric functions, remembering that secant is the reciprocal of cosine can simplify your calculations.

• If the cosine of an angle is given, flip that value to find the secant.
• If the cosine of an angle is zero, then the secant is undefined.

By knowing this, you can easily calculate the secant function for any angle, provided you know or can determine the cosine for that angle.

Unit Circle

The unit circle is a powerful tool in trigonometry, particularly useful when dealing with trigonometric functions. It is a circle with its center at the origin \(0,0\) and a radius of one unit. This simplicity allows for straightforward calculation of all trigonometric functions.

The unit circle helps us understand angles and their corresponding points.

• An angle is measured in radians or degrees and determines how far around the circle the point will be, starting from the positive x-axis.
• Each point on the unit circle corresponds to the x and y coordinates which are cosine and sine of that angle, respectively.

In our specific example of 270 degrees, the unit circle tells us it points straight down, giving the coordinates \( (0, -1) \). The x-coordinate is \(0\), which corresponds to the cosine value, which is crucial for determining secant.

Reciprocal Identities

Reciprocal identities in trigonometry are useful relationships that help simplify calculations and understand the properties of trigonometric functions. One of the main reciprocal identities is that the secant function is the reciprocal of the cosine function, as mentioned: \( \sec \theta = \frac{1}{\cos \theta} \).

Understanding this identity clarifies why certain angles make the secant undefined.

• For angles where the cosine is zero, attempting to calculate the secant means dividing by zero, leading to an undefined result.
• These situations occur at specific points on the unit circle, such as \(90^{\circ}\), \(270^{\circ}\), etc., where the x-value (cosine) for the points on the unit circle are zero.

These reciprocal identities provide a foundation for simplifying more complex trigonometric expressions and solving equations effectively.