Graph of arcsin

Definition

The graph of arcsin, or the inverse sine function, represents the relationship between an angle and its sine value. This function takes a value between -1 and 1 and returns an angle in radians from the range of $$[-\frac{\pi}{2}, \frac{\pi}{2}]$$. Understanding this graph is essential for solving equations involving inverse trigonometric functions, as it helps visualize how angles correspond to their sine values.

5 Must Know Facts For Your Next Test

1. The graph of arcsin is a curve that starts at $$(-1, -\frac{\pi}{2})$$ and ends at $$(1, \frac{\pi}{2})$$.
2. The graph is increasing, meaning as the input (sine value) goes up, the output (angle) also increases.
3. It has a range limited to angles between $$-\frac{\pi}{2}$$ and $$\frac{\pi}{2}$$ because these are the principal values for the inverse sine function.
4. The x-axis intercept occurs at $$0$$, which corresponds to an arcsin value of $$0$$.
5. The graph is symmetric about the origin, indicating that $$arcsin(-x) = -arcsin(x)$$ for all x in the domain.

Review Questions

• How does understanding the graph of arcsin help in solving equations involving inverse trigonometric functions?
  • Understanding the graph of arcsin allows for visualizing how angles correspond to their sine values, which is crucial for solving equations. For instance, when given a sine value, one can determine which angle corresponds to that value by locating it on the graph. This graphical insight aids in grasping not just solutions but also understanding how those solutions behave as inputs change.
• Discuss how the characteristics of the graph of arcsin influence its range and domain in relation to other inverse trigonometric functions.
  • The graph of arcsin shows that its domain is restricted to values between -1 and 1 due to how sine values behave. The corresponding range from $$-\frac{\pi}{2}$$ to $$\frac{\pi}{2}$$ ensures that each sine value has a unique angle output. This contrast with other inverse functions like arccos and arctan highlights differing domains and ranges and emphasizes how each function manages its respective outputs based on their original sine or cosine behaviors.
• Evaluate how the symmetry observed in the graph of arcsin can be applied when solving complex trigonometric equations.
  • The symmetry of the graph of arcsin about the origin allows for quick evaluation when solving complex trigonometric equations. For example, if you find that arcsin(x) = y, then you immediately know that arcsin(-x) = -y. This property simplifies calculations and can significantly reduce solving time by leveraging this symmetry in finding additional solutions across different quadrants without having to redraw or reevaluate parts of the graph.

Related terms

sine function: The sine function relates an angle in a right triangle to the ratio of the length of the opposite side to the length of the hypotenuse.

arccos: The arccosine function is the inverse of the cosine function, which returns an angle whose cosine is a given number.

arctan: The arctangent function is the inverse of the tangent function, returning an angle whose tangent is a given number.