91Ó°ÊÓ

The cotangent function, denoted as \( \cot x \), is the reciprocal of the tangent function and is defined as \( \cot x = \frac{1}{\tan x} \). It is an important function in trigonometry, primarily utilized in problems involving right-angled triangles and periodic functions.
Understanding the inverse, \( \cot^{-1} x \), allows us to find an angle given the cotangent ratio. The range of \( \cot^{-1} x \) is restricted to the interval \( 0 < x < \pi \), ensuring it provides unique angle solutions.

To express \( \cot^{-1} x \) in terms of \( \tan^{-1} \), we use the relationship between cotangent and tangent. Here's how it works:

• When \( x > 0 \), we have \( \cot^{-1} x = \tan^{-1} (1/x) \). This is because \( \cot \) being a reciprocal function translates the problem into an inverse tangent by taking \( 1/x \).
• For \( x < 0 \), periodicity changes the scenario, and the expression becomes \( \cot^{-1} x = \pi + \tan^{-1} (1/x) \). This adjustment accounts for the angle's position within the interval \( 0 < x < \pi \).

By understanding these properties, you can solve inverse cotangent problems with confidence.

The cosecant function, represented as \( \csc x \), is the reciprocal of the sine function, defined by \( \csc x = \frac{1}{\sin x} \). It is particularly useful in trigonometric problems involving reciprocals and identities.
The inverse function, \( \csc^{-1} x \), helps determine angles from the cosecant value. Its range is constrained between \(-\pi/2 < x < \pi/2, x eq 0\) to provide a unique solution.

For the expression \( \csc^{-1} x = \sin^{-1} \left(\frac{1}{x}\right) \) when \(|x| \geq 1\), here's a simple breakdown:

• The relationship \( \csc x = \frac{1}{\sin x} \) allows us to solve for \( \sin x \) in terms of \( \frac{1}{x} \).
• This means \( \sin y = 1/x \), leading to \( \csc^{-1} x = \sin^{-1} (1/x) \).

The interval of \( \csc^{-1} x \) ensures that the angle remains within the accepted range, providing accurate inverse trigonometric solutions.

The secant function, denoted \( \sec x \), is another reciprocal function related to cosine, defined by \( \sec x = \frac{1}{\cos x} \). This function appears frequently in problems relating to angular measurement and trigonometric identities.
When dealing with its inverse, \( \sec^{-1} x \), we find the angle corresponding to a given secant value. The range for this is defined by the condition \( |x| \geq 1 \) to maintain unique and valid solutions.

By using the expression \( \sec^{-1} x = \cos^{-1} \left( \frac{1}{x} \right) \), here’s how we approach it:

• The identity \( \sec y = \frac{1}{\cos y} \) directly implies \( x = \frac{1}{\cos y} \).
• Therefore, solving for \( \cos y \) gives us \( \cos y = 1/x \), which is expressed in inverse terms as \( \sec^{-1} x = \cos^{-1} (1/x) \).

This setup maintains the true values within the domain and range required for correct inverse results.