Graph of arcsine

Definition

The graph of arcsine is a visual representation of the inverse sine function, denoted as $$y = \arcsin(x)$$, which gives the angle whose sine is x. This graph maps the range of the arcsine function, which is limited to angles between $$-\frac{\pi}{2}$$ and $$\frac{\pi}{2}$$ radians (or -90° to 90°), making it distinct from the regular sine function that extends infinitely in both positive and negative directions. The graph has a specific shape that reflects the properties of inverse functions, including being one-to-one and having a restricted domain from -1 to 1, since sine values fall within this interval.

5 Must Know Facts For Your Next Test

1. The domain of the graph of arcsine is restricted to x values between -1 and 1, reflecting the range of the sine function.
2. The graph is a continuous curve that increases from $$-\frac{\pi}{2}$$ to $$\frac{\pi}{2}$$ as x moves from -1 to 1.
3. At the point where x equals 0, the value of arcsine is also 0, which means it intersects the origin (0,0) on the graph.
4. The slope of the arcsine graph varies, being steepest near the endpoints (-1 and 1) and flattening out around 0.
5. The graph exhibits symmetry about the origin, reflecting its property that $$\arcsin(-x) = -\arcsin(x)$$.

Review Questions

• How does the graph of arcsine differ from that of the sine function in terms of range and domain?
  • The graph of arcsine is distinct from that of the sine function because its domain is restricted to x values between -1 and 1, while its range spans from $$-\frac{\pi}{2}$$ to $$\frac{\pi}{2}$$. In contrast, the sine function can take any angle as input but only outputs values between -1 and 1. This restriction ensures that arcsine is one-to-one, allowing it to have an inverse that can be graphed.
• Explain why the graph of arcsine has a unique shape compared to other inverse trigonometric functions.
  • The graph of arcsine has a unique shape due to its restricted domain and range, which creates a continuous increasing curve from $$-\frac{\pi}{2}$$ to $$\frac{\pi}{2}$$. This shape reflects its definition as an inverse function; it essentially 'undoes' the sine function's output. The slopes vary along different segments, being steepest at both ends near -1 and 1 but flattening near zero, showcasing how angles map back from sine values.
• Critically analyze how understanding the graph of arcsine can enhance your comprehension of trigonometric identities and their applications.
  • Understanding the graph of arcsine enhances comprehension of trigonometric identities because it visually demonstrates how angles correspond to their sine values. This connection helps in solving equations involving inverse trigonometric functions. Moreover, recognizing how this graph interacts with other graphs like those of cosine or tangent can reveal deeper insights into relationships among these functions. A thorough grasp of these concepts allows for more advanced problem-solving techniques and application in real-world scenarios such as physics and engineering.

Related terms

sine function: A trigonometric function that relates the angle of a right triangle to the ratio of the length of the opposite side to the hypotenuse.

inverse functions: Functions that reverse the effect of another function; for example, the arcsine function reverses the sine function.

unit circle: A circle with a radius of one centered at the origin of a coordinate plane, used to define trigonometric functions and their inverses.