The exact answer to the question "What is the range of sin theta?" is the interval [-1, 1].
What is the Range of Sin Theta?
The range of a function defines the complete set of all possible output values that the function can produce. For the sine function, denoted as sin θ, its values consistently vary between a minimum of -1 and a maximum of 1, including these two boundary values. This implies that for any real number input for θ, the output of sin θ will always be a value between -1 and 1, inclusive.
This fundamental property is crucial for understanding the behavior of the sine wave, which graphically illustrates this periodic oscillation between its maximum and minimum values.
Domain and Range of Key Trigonometric Functions
To provide a broader context, here is a table outlining the domains (possible input values) and ranges (possible output values) for the primary trigonometric functions:
| Function | Domain (Input Values) | Range (Output Values) |
|---|---|---|
| sin θ | All real numbers (R) or (-∞, ∞) | [-1, 1] |
| cos θ | All real numbers (R) or (-∞, ∞) | [-1, 1] |
| tan θ | All real numbers except odd multiples of π/2 (R - {(2n + 1)π/2 | n ∈ Z}) |
| cot θ | All real numbers except integer multiples of π (R - {nπ | n ∈ Z}) |
As observed in the table, the range of sin θ is distinctly defined as [-1, 1], emphasizing that its output values are bounded and will never fall outside this specific closed interval.
