Step-by-Step Solution

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