[0, π] is the interval on the real number line that includes all real numbers from 0 to π, including 0 and π. This interval is commonly used in the context of inverse trigonometric functions, as it represents the principal domain of these functions.
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The interval $[0, π]$ is the principal domain for the inverse sine, inverse cosine, and inverse tangent functions.
This interval represents all possible angles from 0 radians (0 degrees) to $π$ radians (180 degrees).
The inverse trigonometric functions map the values of the standard trigonometric functions within the interval $[0, π]$ to a unique angle.
The inverse sine function, $\sin^{-1}(x)$, maps the values of sine from -1 to 1 to the angles in the interval $[0, π]$.
The inverse cosine function, $\cos^{-1}(x)$, maps the values of cosine from -1 to 1 to the angles in the interval $[0, π]$.
Review Questions
Explain the significance of the interval $[0, π]$ in the context of inverse trigonometric functions.
The interval $[0, π]$ is the principal domain for the inverse sine, inverse cosine, and inverse tangent functions. This means that the values of the standard trigonometric functions (sine, cosine, and tangent) within this interval are mapped to a unique angle. The inverse trigonometric functions allow you to find the angle given the value of the trigonometric function, and the $[0, π]$ interval represents all possible angles from 0 to 180 degrees, or 0 to $π$ radians.
How do the inverse trigonometric functions utilize the $[0, π]$ interval to determine the unique angle corresponding to a given trigonometric function value?
The inverse trigonometric functions, such as $\sin^{-1}(x)$, $\cos^{-1}(x)$, and $\tan^{-1}(x)$, map the values of the standard trigonometric functions (sine, cosine, and tangent) within the interval $[0, π]$ to a unique angle. This means that for any value of the trigonometric function between -1 and 1, there is a single angle in the $[0, π]$ interval that corresponds to that value. The inverse functions allow you to find this unique angle, which is crucial for solving problems involving inverse trigonometric functions.
Explain the relationship between the $[0, π]$ interval, radian measure, and the behavior of inverse trigonometric functions.
The interval $[0, π]$ represents half a revolution around a circle, or $π$ radians. This interval is significant in the context of inverse trigonometric functions because it represents the principal domain of these functions. The inverse trigonometric functions map the values of the standard trigonometric functions within this interval to a unique angle, expressed in radians. This allows for the determination of the angle given the value of the trigonometric function, which is a fundamental concept in solving problems involving inverse trigonometric functions. The connection between the $[0, π]$ interval, radian measure, and the behavior of inverse trigonometric functions is crucial for understanding and applying these functions effectively.
Inverse trigonometric functions are the inverse of the standard trigonometric functions (sine, cosine, tangent, etc.). They allow you to find the angle given the value of the trigonometric function.
Principal Domain: The principal domain of an inverse trigonometric function is the interval on the real number line where the function is defined and has a unique, single-valued output.
Radian measure is a way of measuring angles, where one full revolution around a circle is equal to $2\pi$ radians. The interval $[0, π]$ represents half a revolution, or $π$ radians.