Problem 223 - Precalculus

Question

For the following exercises, use reference angles to evaluate the expression. If \(\sec t=3.1, \quad\) what is the \(\sec (-t) ?\)

Step-by-Step Solution

Answer

\( \sec(-t) = 3.1 \)

Step 1: Understanding the Properties of Secant

The secant function, \( ext{sec}\), is related to cosine by the identity \( ext{sec} \ heta = \frac{1}{ ext{cos} \ heta}\). An important property of the cosine function is that it is an even function. This means \( ext{cos}(-\theta) = ext{cos}(\theta)\). Thus, \( ext{sec}(-\theta) = ext{sec}(\theta)\) for any angle \( \theta \).

Step 2: Evaluate \\( ext{sec}(-t)\\)

Given that \( ext{sec}(t) = 3.1\), and knowing the property that secant is an even function, it follows that \( ext{sec}(-t) = ext{sec}(t)\). Therefore, \( ext{sec}(-t) = 3.1\).

Key Concepts

Secant Function

The secant function, denoted as \(\sec\), is one of the six fundamental trigonometric functions. It is closely related to the cosine function. In fact, secant is the reciprocal of cosine, which can be expressed as:

• \(\sec \theta = \frac{1}{\cos \theta}\)

This means if you can find the cosine of an angle, simply take its reciprocal to find the secant.
The secant function is typically used less frequently in basic trigonometry, but plays a vital role in more advanced areas of mathematics, like calculus and complex analysis. Remembering that secant goes hand in hand with cosine can greatly simplify solving trigonometric expressions involving secant.Exploring secant's properties and relationships can be very useful, particularly in complex problems where reciprocal relationships help in simplification.

Even Function Properties

An even function is characterized by the fact that it remains unchanged if you reverse the sign of its input. Mathematically, a function \(f(x)\) is termed "even" if it satisfies the condition:

• \(f(-x) = f(x)\).

The cosine function, and therefore the secant function, is regarded as even.
The property of being an even function implies that when you calculate secant of a negative angle, \(\sec(-\theta)\), it will be the same as \(\sec(\theta)\).
This characteristic is particularly useful because it allows you to evaluate angles that might go beyond typical angle boundaries by leveraging the properties of symmetry. In practical terms, this simplifies calculations greatly, enabling easier and more efficient problem-solving in trigonometry and calculus.

Reference Angles

Reference angles are a crucial concept in trigonometry. They help simplify the process of finding trigonometric functions of any angle. A reference angle is always between \(0\) and \(90\) degrees (or \(0\) and \(\pi/2\) radians).
It is essentially the acute angle formed by the terminal side of an angle and the horizontal axis.

• You can derive reference angles by subtracting the relevant quadrant boundaries from your given angle.

The usefulness of reference angles lies in their ability to transform a wide range of angles to a simpler, standard form often easily managed using the common values of trigonometric functions.
To find the reference angle in different quadrants:

• In Quadrant I, the reference angle is the angle itself.
• In Quadrant II, subtract the angle from \(180^\circ\) (or \(\pi\)).
• In Quadrant III, subtract \(180^\circ\) (or \(\pi\)) from the angle.
• In Quadrant IV, subtract the angle from \(360^\circ\) (or \(2\pi\)).

By using the concept of reference angles, you can effectively evaluate trigonometric functions like secant over any angle within the circle.