Problem 27 - Algebra 2 and Trigonometry

Question

For what values of \(\theta\) is the identity \(\frac{\cos \theta}{\sec \theta}+\frac{\sin \theta}{\csc \theta}=1\) undefined?

Step-by-Step Solution

Answer

The identity is undefined when \(\theta = k\pi\) or \(\theta = \frac{\pi}{2} + k\pi\) for integer \(k\).

Step 1: Convert Secant and Cosecant to Cosine and Sine

First, recall the definitions of secant and cosecant. The secant is the reciprocal of cosine: \( ext{sec} \, \theta = \frac{1}{\cos \theta}\). The cosecant is the reciprocal of sine: \(\text{csc} \, \theta = \frac{1}{\sin \theta}\). Substitute these identities into the given expression.

Step 2: Simplify the Expression

Substitute the reciprocal identities: \(\frac{\cos \theta}{\frac{1}{\cos \theta}} + \frac{\sin \theta}{\frac{1}{\sin \theta}} = 1\). Simplify the expression: \(\cos^2 \theta + \sin^2 \theta = 1\). This is always true according to the Pythagorean identity, so the expression simplifies to \(1 = 1\).

Step 3: Identify Undefined Conditions

The identity will be undefined when either the original expression’s denominators \(\sec \theta\) or \(\csc \theta\) are zero. This happens when the reciprocal functions’ numerators are zero, which occurs if \(\cos \theta = 0\) or \(\sin \theta = 0\).

Step 4: Determine Problematic Angles

\(\cos \theta = 0\) occurs when \(\theta = \frac{\pi}{2} + k\pi\) for any integer \(k\). Similarly, \(\sin \theta = 0\) occurs when \(\theta = k\pi\) for any integer \(k\). Analyze these angles to determine where the secant or cosecant functions are undefined.

Key Concepts

Secant and Cosecant Functions

In trigonometry, secant (\(\text{sec} \, \theta\)) and cosecant (\(\text{csc} \, \theta\)) functions play critical roles as they are the reciprocal functions of cosine and sine, respectively. Understanding these functions is essential for solving trigonometric expressions efficiently.

• Secant is defined as the reciprocal of the cosine function: \(\text{sec} \, \theta = \frac{1}{\cos \theta}\). Hence, it involves the same angles where cosine is used but flipped over its axis.
• Cosecant is defined as the reciprocal of the sine function: \(\text{csc} \, \theta = \frac{1}{\sin \theta}\). This function is crucial when working with angles in situations where sine values are very small.

Whenever you encounter these functions, it may be easier to think of them in terms of their basic sine and cosine components. This insight simplifies complex trigonometric equations by allowing you to substitute secant and cosecant with their respective reciprocal identities.

Undefined Trigonometric Expressions

Not all values of \(\theta\) will yield defined results, especially when working in trigonometry. Expressions involving reciprocals can become undefined when their denominators are zero.
For instance, when dealing with secant and cosecant, the following scenarios lead to undefined expressions:

• If \(\cos \theta = 0\), then \(\text{sec} \, \theta = \frac{1}{\cos \theta}\) becomes undefined as division by zero is not possible.
• Similarly, if \(\sin \theta = 0\), then \(\text{csc} \, \theta = \frac{1}{\sin \theta}\) is undefined for the same reason.

By identifying these problematic angles, you can better analyze trigonometric equations and avoid undefined solutions. Always be mindful of these angles when solving equations to ensure your answers are valid and logical.

Pythagorean Identity

A cornerstone in trigonometry is the Pythagorean identity: \(\cos^2 \theta + \sin^2 \theta = 1\). This identity is always true and forms the basis for many trigonometric proofs and solutions.
The identity derives from the Pythagorean theorem, relating to a right-angled triangle where the sum of the squares of the sine and cosine of any angle \(\theta\) equals one. This can be visualized in a unit circle, where any point on the circle satisfies this relationship.

• When solving trigonometric problems, the Pythagorean identity can often simplify complex expressions by transforming the expression into a recognizable form.
• For example, an expression like \(\cos^2 \theta + \sin^2 \theta = 1\) is naturally simplified using the identity, directly resulting in a true statement.

Understanding and leveraging the Pythagorean identity can significantly streamline your work with trigonometric equations, making it a vital tool in your mathematical toolkit.