Problem 3 - Precalculus with Limits

Question

In Exercises 3-8, fill in the blank to complete the trigonometric identity. \( \dfrac{1}{\cot u} \)= ________

Step-by-Step Solution

Answer

\( \tan u \)

Step 1: Identify the trigonometric function

The equation is given in terms of cotangent of u, which is denoted as \( \cot u \). Now the goal is to find the function on the right side of the equation that is equal to \( \dfrac{1}{\cot u} \).

Step 2: Recall the reciprocal property

By the properties of trigonometric functions, the cotangent function is defined as the reciprocal of the tangent function. Thus, \( \cot u = \dfrac{1}{\tan u} \).

Step 3: Apply the reciprocal property

Applying this property to the given equation, \( \dfrac{1}{\cot u} \) becomes \( \dfrac{1}{\dfrac{1}{\tan u}} \).

Step 4: Simplify the expression

Upon simplifying, the expression becomes \( \tan u \), which completes the trigonometric identity.

Key Concepts

Cotangent Function

The cotangent function is an essential trigonometric function, often denoted as \( \text{cot} \). It is defined as the ratio of the adjacent side to the opposite side in a right-angled triangle. However, for an angle \( u \), the cotangent function is more commonly expressed in terms of sine and cosine functions.

The cotangent of an angle is the reciprocal of its tangent, which can be mathematically shown as:\[ \text{cot} u = \frac{1}{\text{tan} u} \]or alternatively as the ratio of the cosine to the sine:\[ \text{cot} u = \frac{\text{cos} u}{\text{sin} u} \]Understanding the cotangent function is not only key in solving trigonometric identities but also in various applications such as calculus, physics, and engineering. When you encounter an expression involving \( \frac{1}{\text{cot} u} \), remember it represents the tangent function based on its reciprocal property.

Reciprocal Property of Trigonometry

In trigonometry, the reciprocal property is a fundamental characteristic that relates certain trigonometric functions to one another. Particularly, it states that the tangent, cotangent, secant, and cosecant functions are the reciprocals of one another in specific pairs.

• The reciprocal of the tangent function is the cotangent function: \( \text{tan} u = \frac{1}{\text{cot} u} \)
• The reciprocal of the secant function is the cosine function: \( \text{sec} u = \frac{1}{\text{cos} u} \)
• The reciprocal of the cosecant function is the sine function: \( \text{csc} u = \frac{1}{\text{sin} u} \)

These relationships are crucial in simplifying complex trigonometric expressions and solving equations. By applying the reciprocal property, you can often transform an unfamiliar trigonometric term into a more familiar one, making the problem easier to tackle.

Tangent Function

The tangent function, symbolized as \( \text{tan} \), is another primary trigonometric function. It is defined for an angle \( u \) as the ratio of the sine function to the cosine function, that is,\[ \text{tan} u = \frac{\text{sin} u}{\text{cos} u} \]In the context of a right triangle, it represents the ratio of the length of the opposite side to the length of the adjacent side. The tangent function has a period of \( \text{π} \) radians (or 180 degrees), meaning it repeats its values every \( \text{π} \) radians. An essential aspect of the tangent function is its relationship with circles, particularly the unit circle, where the tangent of an angle can be visually interpreted as the length of the segment that extends from the origin to where the line, at the given angle, intersects the tangent to the unit circle at the point (1,0).When you come across \( \frac{1}{\text{cot} u} \) in trigonometric identities or equations, you can now confidently replace it with \( \text{tan} u \) by using the reciprocal property discussed. This property simplifies the process of solving trigonometric problems by recognizing the inherent relationships between the functions.