Question

CRITICAL THINKING Write \(\tan \theta\) as the ratio of two other trigonometric functions. Use this ratio to explain why \(\tan 90^{\circ}\) is undefined but \(\cot 90^{\circ}=0\).

Short answer

The tangent function \( \tan \theta \) can be expressed as the ratio of sine over cosine \( \sin \theta / \cos \theta \). This reveals that \( \tan 90^{\circ} \) is undefined because it involves division by zero. On the other hand, \( \cot 90^{\circ} \) equals zero, because cotangent is the reciprocal of tangent and a number divided by an undefined value is zero.

Step-by-step solution

Express \( \tan \theta \) As a Ratio of Other Trigonometric Functions

Firstly, use the identity \( \tan \theta = \frac{\sin \theta}{\cos \theta} \) to express the function \( \tan \theta \) as a ratio of other trigonometric functions.

Identify Why \( \tan 90^{\circ} \) is Undefined

Next, substitute \( \theta = 90^{\circ} \) into the identity from Step 1. Thus, \( \tan 90^{\circ} = \frac{\sin 90^{\circ}}{\cos 90^{\circ}} = \frac{1}{0} \). Since division by zero is undefined in mathematics, it is clear why \( \tan 90^{\circ} \) is undefined.

Identify Why \( \cot 90^{\circ} = 0 \)

Finally, since the cotangent of an angle is by definition the reciprocal of the tangent of the angle, \( \cot \theta = \frac{1}{\tan \theta} \). Thus, substituting \( \theta = 90^{\circ} \) into the cotangent relationship provides \( \cot 90^{\circ} = \frac{1}{\tan 90^{\circ}} = \frac{1}{\text{undefined}} = 0 \).

Key concepts

Understanding the Tangent Function

The tangent function, denoted as \( \tan(\theta) \), is one of the core trigonometric functions in mathematics. It is most commonly expressed as the ratio of two other important trigonometric functions: the sine function (\( \sin(\theta) \)) and the cosine function (\( \cos(\theta) \)).
\[ \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \]
This definition helps us understand the behavior of the tangent function at various angles. The tangent of an angle in a right triangle can also be seen as the ratio of the opposite side to the adjacent side. This makes \( \tan(\theta) \) a useful measure for understanding slope or steepness at a given angle.
When exploring the properties and values of \( \tan(\theta) \), it's crucial to remember that the behavior of each sine and cosine function can directly influence the value of \( \tan(\theta) \). This relationship is especially evident at certain key angles, like \( 90^\circ \), where it is necessary to consider the concept of undefined expressions.

The Role of Cotangent Function

The cotangent function, symbolized as \( \cot(\theta) \), is another essential trigonometric function that is closely related to the tangent function. By definition, it is the reciprocal of the tangent function:
\[ \cot(\theta) = \frac{1}{\tan(\theta)} = \frac{\cos(\theta)}{\sin(\theta)} \]
This relationship between \( \cot(\theta) \) and \( \tan(\theta) \) allows us to calculate cotangent values using sine and cosine. At certain angles, the value of \( \cot(\theta) \) shows dramatic differences compared to \( \tan(\theta) \).
For example, at \( 90^\circ \), while \( \tan(90^\circ) \) is undefined due to division by zero, \( \cot(90^\circ) \) equals zero. This happens because the roles of \( \sin \) and \( \cos \) are reversed, resulting in
\[ \cot(90^\circ) = \frac{1}{\tan(90^\circ)} \]
Having an intuitive understanding of cotangent provides a complete view of trigonometric ratios and aids in solving complex trigonometric equations.

Significance of Undefined Expressions in Trigonometry

Undefined expressions, such as dividing by zero, often arise in trigonometry, primarily when dealing with tangent and cotangent functions. An undefined expression indicates that a value cannot be determined within the standard real number system.
One common example of an undefined expression is \( \tan(90^\circ) \), which results from \( \cos(90^\circ) = 0 \) leading to division by zero in the ratio \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \). Because division by zero is mathematically undefined, \( \tan(90^\circ) \) remains undefined.
Moreover, these undefined situations provide rational checks and are critical for understanding the periodic and discontinuous nature of trigonometric functions. When performing mathematical operations, always watch for break points that could result in undefined expressions. This awareness ensures that solutions remain valid and correct, preserving the integrity of mathematical conclusions.