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Chapter 5: Problem 3

Fill in the blank to complete the trigonometric identity. \(\frac{1}{\tan u}=\)________

Short Answer

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\(\cot u\)

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01

Understanding Reciprocal Trigonometric Identities

Reciprocal trigonometric identities are those where each trigonometric function has a reciprocal, which is simply one divided by the function itself. For instance, the reciprocal of sine is cosecant, the reciprocal of cosine is secant, and the reciprocal of tangent is cotangent.
02

Finding the Reciprocal of Tangent

In this exercise we need to find the reciprocal of the tangent function. Its reciprocal is the cotangent of u, which is simply written as \(\cot u \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Reciprocal Trigonometric Identities
Reciprocal trigonometric identities are fundamental concepts in trigonometry. Each primary trigonometric function has a corresponding reciprocal function.
This means that you take one divided by the main function to get its reciprocal.
These identities allow you to explore the relationships between different trigonometric functions, making them very useful in various mathematical contexts.Some examples of reciprocal trigonometric identities include:
  • The reciprocal of sine ( \( \sin u \) ) is cosecant ( \( \csc u = \frac{1}{\sin u} \) ).
  • The reciprocal of cosine ( \( \cos u \) ) is secant ( \( \sec u = \frac{1}{\cos u} \) ).
  • The reciprocal of tangent ( \( \tan u \) ) is cotangent ( \( \cot u = \frac{1}{\tan u} \) ).
Understanding these reciprocals is key to solving more complex trigonometric problems, as they provide versatile tools for manipulation and simplification.
Tangent Function
The tangent function, often abbreviated as tan, is one of the core trigonometric functions.
It can be defined as the ratio of the sine and cosine functions. In mathematical terms, it is expressed as:\[\tan u = \frac{\sin u}{\cos u}\] This equation shows that tangent represents the relationship between sine and cosine.
In a right-angled triangle, the tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side.
Some key points about the tangent function:
  • Tangent is periodic with a period of \(\pi\).
  • The function is undefined when \(\cos u = 0\).
  • Tangent can take any value from negative to positive infinity.
Knowing how tangent functions behave and how they are defined is crucial for understanding their role in trigonometry and calculus.
Cotangent
The cotangent function, represented as \(\cot u\), is the reciprocal of the tangent function.
It gives a way to represent the same trigonometric relationship in a different form. You can calculate cotangent using the formula:\[\cot u = \frac{1}{\tan u} = \frac{\cos u}{\sin u}\]This shows that cotangent is the inverse of tangent, both multiplicatively and in terms of its ratio with cosine and sine.
In a right triangle, cotangent is the ratio of the adjacent side to the opposite side.Some characteristics of the cotangent function include:
  • Cotangent has a period of \(\pi\).
  • It is undefined when \(\sin u = 0\).
  • Like tangent, cotangent can range from negative to positive infinity.
Understanding cotangent helps in solving trigonometric equations and in applications involving periodic functions.

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