Question

How to find the range and domain of any trigonometric ratio.

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Answer

To determine the domain and range of a trigonometric function, we can follow these steps:

  1. Understanding the Domain:

    • Domain is the set of input values (on the x-axis) for which the function is defined.
    • For most trigonometric functions, the domain consists of angles, usually represented by $ \theta $.
  2. Understanding the Range:

    • Range is the set of output values (on the y-axis) that the function can produce.

Example:

Let's find the domain and range for the function $ \sin(\theta) $.

  • Domain: For $ \sin(\theta) $, the angle $ \theta $ can vary from $0^\circ$ to $360^\circ$ (or equivalently from $0$ to $2\pi$ radians). Therefore, the domain of $ \sin(\theta) $ is all real numbers.

  • Range: The sine function $ \sin(\theta) $ outputs values between $ -1 $ and $ 1 $. Hence, the range of $ \sin(\theta) $ is: $$ -1 \leq \sin(\theta) \leq 1 $$

In summary:

  • Domain: Inputs or $ \theta $, which spans typically all real numbers for standard trigonometric functions.
  • Range: Outputs dependent on the specific trigonometric function, e.g., for $ \sin(\theta) $, it is $ [-1, 1] $.

By understanding these principles, you can determine the domain and range for any trigonometric ratio.


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