Inverse Trigonometric Functions Calculator
Calculate inverse trigonometric functions (arcsin, arccos, arctan) and their values.
Inverse Trigonometric Functions
The inverse trigonometric functions are defined as:
- \[ \arcsin(x) = \theta \text{ where } \sin(\theta) = x \text{ and } -\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2} \]
- \[ \arccos(x) = \theta \text{ where } \cos(\theta) = x \text{ and } 0 \leq \theta \leq \pi \]
- \[ \arctan(x) = \theta \text{ where } \tan(\theta) = x \text{ and } -\frac{\pi}{2} < \theta < \frac{\pi}{2} \]
Properties:
- Domain of arcsin: [-1, 1]
- Domain of arccos: [-1, 1]
- Domain of arctan: (-∞, ∞)
- Range of arcsin: [-π/2, π/2]
- Range of arccos: [0, π]
- Range of arctan: (-π/2, π/2)
Common Values:
- \(\arcsin(0) = 0\)
- \(\arcsin(\frac{1}{2}) = \frac{\pi}{6}\)
- \(\arcsin(\frac{1}{\sqrt{2}}) = \frac{\pi}{4}\)
- \(\arcsin(\frac{\sqrt{3}}{2}) = \frac{\pi}{3}\)
- \(\arccos(0) = \frac{\pi}{2}\)
- \(\arccos(\frac{1}{2}) = \frac{\pi}{3}\)
- \(\arctan(0) = 0\)
- \(\arctan(1) = \frac{\pi}{4}\)
Applications:
- Trigonometry
- Calculus (integration)
- Physics (wave motion)
- Engineering (signal processing)
- Computer graphics
- Navigation and surveying
Geometric Interpretation:
- arcsin(x): Angle whose sine is x
- arccos(x): Angle whose cosine is x
- arctan(x): Angle whose tangent is x
- Used to find angles in right triangles
- Important in solving trigonometric equations