Problem 49 - Algebra 2

Question

Explain why the equation \(\sec \theta=0\) has no solutions.

Step-by-Step Solution

Answer

The equation has no solutions because \(\frac{1}{\cos \theta} = 0\) is undefined.

Step 1: Understanding the Secant Function

The secant function, represented as \(\sec \theta\), is the reciprocal of the cosine function. This means \(\sec \theta = \frac{1}{\cos \theta}\). For \(\sec \theta\) to equal zero, \(\frac{1}{\cos \theta} = 0\) must be true.

Step 2: Analyzing the Reciprocal Property

The expression \(\frac{1}{\cos \theta}\) can only be zero if \(\cos \theta\) is undefined or does not exist. However, a reciprocal of zero implies division by zero, which is undefined in mathematics.

Step 3: Exploring Cosine Values

Since \(\cos \theta\) can never be zero for \(\sec \theta\) to exist, we know that there are no values of \(\theta\) that make \(\cos \theta\) undefined in the real number system.

Step 4: Concluding No Solutions Exist

Since it is impossible for \(\frac{1}{\cos \theta}\) to equal zero, the equation \(\sec \theta = 0\) has no solutions within the set of real numbers.

Key Concepts

Reciprocal of Cosine

The secant function, denoted as \(\sec \theta\), is known for being the reciprocal of the cosine function. Mathematically, we express this relationship as \(\sec \theta = \frac{1}{\cos \theta}\).
This fundamental principle means that the value of \(\sec \theta\) depends on the value of \(\cos \theta\).
In simpler terms, whenever you need to calculate the secant of an angle \(\theta\), you first determine the cosine of that angle and then take its reciprocal.

• If \(\cos \theta\) is a number like 1 or -1, \(\sec \theta\) becomes 1 or -1, respectively.
• However, if \(\cos \theta\) were to be zero, we encounter a mathematical dilemma, as division by zero is undefined.

Understanding this reciprocal relationship helps set the stage for approaching problems involving the secant function, particularly when contemplating potential solutions for expressions like \(\sec \theta = 0\).

Undefined in Mathematics

In mathematics, there is a concept of expressions that are undefined. A common situation where this arises is division by zero.
For any non-zero number \(a\), the expression \(\frac{a}{0}\) does not exist in the realm of real numbers. This leads to an important rule: any solution implying division by zero is considered undefined.

• When working with trigonometric functions like secant, understanding when an expression becomes undefined is crucial.
• If \(\cos \theta\) equals zero, then \(\sec \theta = \frac{1}{\cos \theta}\) can't be computed, as it would require dividing by zero.
• This means that for \(\sec \theta\) to equal zero, it would imply \(\frac{1}{0}\), which is a contradiction and hence not possible.

The idea of undefined expressions helps in identifying impossible solutions and clarifies why certain equations, like \(\sec \theta = 0\), lack solutions in practical scenarios.

Trigonometric Equations

Trigonometric equations are mathematical equations that involve trigonometric functions like sine, cosine, and secant. Solving these equations typically involves identifying the values of the angle \(\theta\) that make the equation true.
A general approach includes considering the properties and graphs of trigonometric functions such as periodicity and amplitude.

• For instance, in the equation \(\sec \theta = 0\), the goal is to find values of \(\theta\) where this condition holds true.
• However, understanding that \(\sec \theta = \frac{1}{\cos \theta}\), the equation can only be solved if \(\frac{1}{\cos \theta}\) truly can equate to zero, which is not feasible given the properties of cosine.
• This renders certain trigonometric equations, like \(\sec \theta = 0\), inherently unsolvable within the real number system.

Approaching trigonometric equations requires not only algebraic manipulation but a profound comprehension of trigonometric principles and their implications on potential solutions.