Problem 46 - Precalculus

Question

Evaluate each expression, if possible. $$\cot 450^{\circ}-\cos \left(-450^{\circ}\right)$$

Step-by-Step Solution

Answer

The expression \(\cot 450^\circ - \cos(-450^\circ)\) is undefined because \(\cot 90^\circ\) is undefined.

Step 1: Determine Equivalent Angle for Cotangent

The given angle for cotangent is \(450^\circ\). Angles greater than \(360^\circ\) can be reduced by subtracting \(360^\circ\) until the angle is in the first rotation. Thus, \(450^\circ - 360^\circ = 90^\circ\). Then, \(\cot 450^\circ = \cot 90^\circ\).

Step 2: Evaluate Cotangent of Equivalent Angle

Recall that \(\cot \theta = \frac{1}{\tan \theta}\). At \(90^\circ\), \(\tan 90^\circ\) is undefined because the sine of \(90^\circ\) is 1 and the cosine of \(90^\circ\) is 0, leading to division by zero. Thus, \(\cot 90^\circ\) is undefined.

Step 3: Determine Equivalent Angle for Cosine

The given angle for cosine is \(-450^\circ\). Angles smaller than \(0^\circ\) can be adjusted by adding \(360^\circ\). Thus, \(-450^\circ + 360^\circ = -90^\circ\) and then another rotation gives \(-90^\circ + 360^\circ = 270^\circ\). So \(\cos(-450^\circ) = \cos 270^\circ\).

Step 4: Evaluate Cosine of Equivalent Angle

At \(270^\circ\), the cosine is 0 because the point at this position on the unit circle is \((0, -1)\). Thus, \(\cos 270^\circ = 0\).

Step 5: Combine Results

Since \(\cot 90^\circ\) is undefined, the expression \(\cot 450^\circ - \cos(-450^\circ)\) does not have a defined value. Therefore, the expression is undefined.

Key Concepts

Understanding Cotangent Evaluation

Cotangent is a trigonometric function often expressed as the reciprocal of the tangent function. For any angle \( \theta \), the identity is given by:

• \( \cot \theta = \frac{1}{\tan \theta} \).

To evaluate \( \cot 450^\circ \), we first convert the angle to an equivalent angle between \( 0^\circ \) and \( 360^\circ \). Any angle greater than \( 360^\circ \) can be reduced by subtracting \( 360^\circ \). Hence, \( 450^\circ - 360^\circ = 90^\circ \). At \( 90^\circ \), the tangent function is undefined because tangent, \( \tan \theta = \frac{\sin \theta}{\cos \theta} \), encounters a division by zero from the cosine of \( 90^\circ \) (which is 0). As a result, \( \cot 90^\circ \) or equivalently \( \cot 450^\circ \) is undefined. Remember:

• Whenever tangent is undefined, cotangent is also undefined.

Hence, when evaluating cotangent at angles like \( 90^\circ \), \( 270^\circ \), avoid division by zero situations.

Exploring Cosine Evaluation

The cosine function evaluates the horizontal coordinate of the unit circle at a given angle. Adjusting angles outside \(0^\circ\) to \(360^\circ\) typically involves adding or subtracting \(360^\circ\) until the angle is within one complete rotation. In our exercise, we examine \( \cos(-450^\circ) \). First, simplify by adding \(360^\circ\), and then, since the angle is still negative, add another \(360^\circ\):

• \( -450^\circ + 360^\circ = -90^\circ \)
• \( -90^\circ + 360^\circ = 270^\circ \)

Thus, \( \cos(-450^\circ) = \cos 270^\circ \). At \(270^\circ\), the cosine value, representing the unit circle's horizontal position at that angle, is zero. Hence:

• \( \cos 270^\circ = 0 \).

Recognize that cosine evaluations directly relate to specific points on the unit circle, often representing key angles like \(0^\circ\), \(90^\circ\), \(180^\circ\), and \(270^\circ\).

The Concept of Undefined Expressions

In trigonometry, an expression is termed "undefined" when it involves a mathematical operation that is not possible within the real number system. Common scenarios for undefined expressions include division by zero or taking root of a negative number without complex numbers.
Such undefined scenarios occur frequently with certain trigonometric functions at specific angles. For example, in the exercise, \( \cot 90^\circ \) is undefined due to division by zero as the tangent function \( \tan 90^\circ \) does not exist. An undefined result for any part of an expression makes the entire expression undefined.
In practical terms:

• An undefined expression signals an impossible calculation within given constraints.
• It's important to recognize these conditions to understand limitations of trigonometric evaluations.

Understanding undefined expressions in trigonometry is vital to correctly interpret function behaviors, ensuring comprehension of possible function limits.