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Chapter 7: Problem 8

Find an equivalent expression for each of the following. $$\csc \left(x+\frac{\pi}{2}\right)$$

Short Answer

Expert verified
\( \sec(x) \)

Step by step solution

01

Understanding the Co-Function Identity

Recall the co-function identity for cosecant: \( \csc(x + \frac{\pi}{2}) = \sec(x) \). Co-function identities relate the trigonometric functions of complementary angles.
02

Applying the Co-Function Identity

Using the co-function identity: \( \csc(x + \frac{\pi}{2}) = \sec(x) \), simplifies the given expression.

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Key Concepts

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

Co-Function Identities
Co-function identities are an essential part of trigonometry. These identities relate the trigonometric functions of complementary angles. Two angles are complementary if their sum is \( \frac{\text{蟺}}{2} \) (which is 90 degrees). The co-function identities for sine, cosine, tangent, cotangent, secant, and cosecant are:
  • \( \text{sin}(\frac{\text{蟺}}{2} - x) = \text{cos}(x) \)
  • \( \text{cos}(\frac{\text{蟺}}{2} - x) = \text{sin}(x) \)
  • \( \text{tan}(\frac{\text{蟺}}{2} - x) = \text{cot}(x) \)
  • \( \text{cot}(\frac{\text{蟺}}{2} - x) = \text{tan}(x) \)
  • \( \text{sec}(\frac{\text{蟺}}{2} - x) = \text{csc}(x) \)
  • \( \text{csc}(\frac{\text{蟺}}{2} - x) = \text{sec}(x) \)
Understanding these can help you transform and simplify expressions easily. For instance, knowing \( \text{csc}(x + \frac{\text{蟺}}{2}) = \text{sec}(x) \) directly simplifies the given problem.
Cosecant (\( \text{csc} \))
Cosecant (abbreviated as \( \text{csc} \)) is one of the six fundamental trigonometric functions. It is the reciprocal of the sine function.
  • \[ \text{csc}(x) = \frac{1}{\text{sin}(x)} \]
  • \[ \text{csc}(x) = \frac{\text{hypotenuse}}{\text{opposite}} \] in a right-angled triangle
Since it's the reciprocal of sine, when \( \text{sin}(x) \) is zero, \( \text{csc}(x) \) is undefined. In terms of the unit circle, \( \text{csc}(x) \) represents the ratio of the radius to the y-coordinate of the point on the circle. In the co-function identities, \( \text{csc}(x + \frac{\text{蟺}}{2}) = \text{sec}(x) \) provides a way to connect \( \text{csc} \) and \( \text{sec} \).
Secant (\( \text{sec} \))
Secant (abbreviated as \( \text{sec} \)) is another fundamental trigonometric function. It is the reciprocal of the cosine function.
  • \[ \text{sec}(x) = \frac{1}{\text{cos}(x)} \]
  • \[ \text{sec}(x) = \frac{\text{hypotenuse}}{\text{adjacent}} \] in a right-angled triangle
Just like \( \text{csc} \), when \( \text{cos}(x) \) is zero, \( \text{sec}(x) \) is undefined. On the unit circle, \( \text{sec}(x) \) represents the ratio of the radius to the x-coordinate of the point on the circle. Connecting to co-function identities, \( \text{csc}(x + \frac{\text{蟺}}{2}) = \text{sec(x)} \) illustrates how the secant function transforms with a phase shift of \( \frac{\text{蟺}}{2} \).

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