Pregunta
Define the reciprocal identity for sine and cosine. How can these identities be represented mathematically?
Ask by Montgomery Harmon. in South Africa
Jan 30,2025
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The reciprocal identities for sine and cosine are:
-
Cosecant (csc) is the reciprocal of sine (sin):
-
Secant (sec) is the reciprocal of cosine (cos):
These identities help in simplifying trigonometric expressions and solving equations by expressing one trigonometric function in terms of another.
Solución
Reciprocal identities are fundamental relationships in trigonometry that express one trigonometric function as the reciprocal of another. Specifically for sine and cosine, the reciprocal identities involve the functions cosecant and secant, respectively. These identities are essential for simplifying trigonometric expressions and solving equations.
Reciprocal Identities for Sine and Cosine
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Cosecant as the Reciprocal of SineThe cosecant function is the reciprocal of the sine function. Mathematically, this relationship is expressed as:
- Domain Consideration: Since division by zero is undefined,
is undefined wherever . This occurs at integer multiples of (i.e., , where is an integer).
- Domain Consideration: Since division by zero is undefined,
-
Secant as the Reciprocal of CosineThe secant function is the reciprocal of the cosine function. This relationship is mathematically represented as:
- Domain Consideration: Similarly,
is undefined wherever . This happens at odd multiples of (i.e., , where is an integer).
- Domain Consideration: Similarly,
Summary of Reciprocal Identities
Function | Reciprocal Identity |
---|---|
|
|
|
|
Example Usage
Suppose you know that
, and you want to find
:
Similarly, if
, then:
These reciprocal identities are particularly useful in various applications, including solving trigonometric equations, integration in calculus, and modeling periodic phenomena in physics and engineering.
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Beyond the Answer
The reciprocal identity for sine states that the sine of an angle is equal to the reciprocal of its cosecant, or mathematically, sin(θ) = 1/csc(θ). Similarly, the reciprocal identity for cosine says that the cosine of an angle is equal to the reciprocal of its secant, represented as cos(θ) = 1/sec(θ). These identities highlight the relationship between these trigonometric functions and their reciprocals, forming a fundamental part of trigonometry.
These identities can be incredibly useful in simplifying expressions or solving equations in trigonometry. For instance, if you encounter a problem involving the cosecant of an angle but only know the sine, you can easily switch between them using the reciprocal identity. It allows for flexibility in problem-solving, turning complicated functions into simpler forms!
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