Problem 342 - Precalculus

Question

For the following exercises, use reference angles to evaluate the given expression. If \(\sec (t)=-2.5,\) what is the \(\sec (-t) ?\)

Step-by-Step Solution

Answer

\( \sec(-t) = -2.5 \).

Step 1: Understanding Secant and Reference Angles

The secant function, denoted as \( \sec(t) \), is the reciprocal of the cosine function: \( \sec(t) = \frac{1}{\cos(t)} \). To find \( \sec(-t) \), we need to understand how the cosine function behaves with negative angles. Cosine is an even function, which means \( \cos(-t) = \cos(t) \). As a result, \( \sec(-t) = \sec(t) \) because the secant function will also satisfy this property.

Step 2: Evaluating \( \sec(-t) \)

Given \( \sec(t) = -2.5 \), since the secant function is even, the evaluation becomes straightforward: \( \sec(-t) = \sec(t) = -2.5 \). Hence, using the even property of both cosine and secant, the secant of the negative angle is the same as the secant of the angle.

Key Concepts

Secant Function

The secant function is a trigonometric function represented by \( \sec(t) \). It is the reciprocal of the cosine function, which means \( \sec(t) = \frac{1}{\cos(t)} \). This signifies that when the cosine of an angle is zero, the secant is undefined, as division by zero is impossible. Understanding this reciprocal relationship is crucial when dealing with trigonometric identities and calculations.
To grasp the concept further, note the following:

• If \( \cos(t) = 0.5 \), then \( \sec(t) = \frac{1}{0.5} = 2 \).
• If \( \cos(t) = -1 \), then \( \sec(t) = -1 \), since it's a direct reciprocal in this instance.
• Secant is undefined for angles where cosine equals zero, such as \( 90^\circ \) or \( 270^\circ \), since division by zero isn't possible.

Thus, understanding secant as the reciprocal function provides an insight into its behavior across different angles.

Reference Angles

Reference angles are useful when evaluating trigonometric functions for any angle. A reference angle is the smallest angle that a given angle makes with the x-axis. It is always between \( 0^\circ \) and \( 90^\circ \) and is noted in degrees, though radians can also be used.
Here are some key points:

• An angle in the first quadrant is its own reference angle.
• For angles in the second quadrant, the reference angle is \( 180^\circ - \text{the angle} \).
• In the third quadrant, subtract \( 180^\circ \) from the angle to find the reference angle.
• For the fourth quadrant, the reference angle is calculated as \( 360^\circ - \text{the angle} \).

Reference angles allow us to use the same trig values we find on the unit circle, regardless of the angle's quadrant, by shifting the problem back to a familiar region in the first quadrant.

Even Functions

Even functions are a special class of functions in mathematics where the output values are symmetric around the y-axis. In simple terms, for any function \( f(x) \) that is even, \( f(-x) = f(x) \). This symmetry implies that the function values are identical no matter if the input value is positive or negative.
When it comes to trigonometric functions, cosine is a well-known even function, meaning \( \cos(-t) = \cos(t) \). The secant function, being \( \sec(t) = \frac{1}{\cos(t)} \), automatically inherits this even nature because the reciprocal doesn't alter the even property:
- \( \sec(-t) = \frac{1}{\cos(-t)} = \frac{1}{\cos(t)} = \sec(t) \).
This property makes solving problems with negative angles straightforward, as it keeps calculations constant regardless of the angle's sign.