In the context of the sine function, '1' represents the maximum value that the sine function can attain. The angles for which sin(x) = 1 are those where the y-coordinate on the unit circle is at its highest point.
Understanding Sine and its Maximum Value
The sine function, written as sin(x), describes the y-coordinate of a point on the unit circle as the angle x changes. The unit circle has a radius of 1, so the y-coordinate ranges from -1 to 1.
- Maximum Value: The sine function reaches its maximum value of 1.
- Location: This occurs when the angle x is 90 degrees (π/2 radians).
Angles Where sin(x) = 1
Since the sine function is periodic, it reaches the value of 1 at multiple angles. These angles are of the form:
x = (π/2) + 2πn, where n is an integer (..., -2, -1, 0, 1, 2, ...).
This means:
- x = π/2 (90 degrees)
- x = π/2 + 2π = 5π/2 (450 degrees)
- x = π/2 + 4π = 9π/2 (810 degrees)
- And so on...
Similarly, we can find negative angles:
- x = π/2 - 2π = -3π/2 (-270 degrees)
- x = π/2 - 4π = -7π/2 (-630 degrees)
- And so on...
Arcsin(1)
The inverse sine function, denoted as arcsin(1) or sin-1(1), asks the question: "What angle has a sine value of 1?" The principal value of arcsin(1) is π/2 (90 degrees). The term "principal value" refers to the value within the defined range of the arcsin function, which is typically between -π/2 and π/2.
| Function | Value | Angle (radians) | Angle (degrees) |
|---|---|---|---|
| sin(π/2) | 1 | π/2 | 90 |
| arcsin(1) | π/2 + 2πn | π/2 | 90 |
Conclusion
In summary, "1 in sin" most accurately describes the maximum value the sine function can achieve. The angles that produce this maximum value are π/2 + 2πn, where n is any integer. Arcsin(1) returns the angle whose sine is 1.
