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The inverse sine function is commonly represented as \( \text{sin}^{-1}(x) \) or \( \text{arcsin}(x) \). This function helps you find the angle whose sine value is x. In order to understand and use this function properly, it is crucial to know its domain.The domain of the inverse sine function includes all input values from \( -1 \) to \( 1 \) inclusive. In mathematical terms, this is written as \( -1 \leq x \leq 1 \). This domain is defined because the sine of an angle, in a right triangle, can only result in values between \( -1 \) and \( 1 \).
Here are a few key points to remember about the inverse sine function:\

• It is used to determine the angle given the sine value.
• The function is only defined between \( -1 \) and \( 1 \).
• Its range is from \( -\frac{\pi}{2} \) to \( \frac{\pi}{2} \).

The inverse cosine function is represented as \( \text{cos}^{-1}(x) \) or \( \text{arccos}(x) \). This function is used to find the angle whose cosine value is x. Similar to the inverse sine function, the inverse cosine function also has a specific domain.The domain of the inverse cosine function spans all values from \( -1 \) to \( 1 \), expressed as \( -1 \leq x \leq 1 \). This is the same range as the cosine function itself, which explains why these values are permissible.
A few important facts about the inverse cosine function:\

• It determines the angle for a given cosine value.
• The function is defined between \( -1 \) and \( 1 \).
• Its range is from \( 0 \) to \( \pi \).

By understanding this domain, you can use the inverse cosine function to solve a wide variety of trigonometric problems.

The inverse tangent function is denoted as \( \text{tan}^{-1}(x) \) or \( \text{arctan}(x) \). This function comes in handy when you need to find the angle whose tangent value is x. Unlike the inverse sine and cosine functions, the inverse tangent function has a different domain.The domain of the inverse tangent function includes all real numbers. This means \( \text{tan}^{-1}(x) \) is defined for all x in the set of real numbers. The reason for this broad domain is that the tangent function can produce any real number as a result.
Some key points about inverse tangent function are

• It identifies the angle given a tangent value.
• The function is defined for all real x values.
• Its range is from \( -\frac{\pi}{2} \) to \( \frac{\pi}{2} \).

This makes the inverse tangent function extremely flexible and useful in trigonometric calculations.