Inverse Trig Calculator

Inverse Trigonometric Functions:

\[ \begin{cases} \arcsin(x) = \theta \\ \arccos(x) = \theta \\ \arctan(x) = \theta \end{cases} \]

Input Value (x):

(must be between -1 and 1 for arcsin/arccos)

Function:

Output Unit:

Result:

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1. What are Inverse Trigonometric Functions?

Inverse trigonometric functions (arcsin, arccos, arctan) calculate the angle whose trigonometric value equals the given number. They are the inverse operations of the basic trigonometric functions.

2. How Does the Calculator Work?

The calculator uses PHP's built-in trigonometric functions:

\[ \begin{cases} \arcsin(x) = \sin^{-1}(x) \\ \arccos(x) = \cos^{-1}(x) \\ \arctan(x) = \tan^{-1}(x) \end{cases} \]

Where:

\( x \) — Input value (must be between -1 and 1 for arcsin/arccos)
Result — Angle in radians or degrees

Explanation: The calculator returns the principal value of the inverse function, with appropriate range restrictions for each function.

3. Understanding the Results

Details:

arcsin(x): Returns angle between -π/2 and π/2 radians (-90° to 90°)
arccos(x): Returns angle between 0 and π radians (0° to 180°)
arctan(x): Returns angle between -π/2 and π/2 radians (-90° to 90°)

4. Using the Calculator

Tips:

For arcsin and arccos, input must be between -1 and 1
For arctan, any real number is valid
Choose whether you want results in radians or degrees

5. Frequently Asked Questions (FAQ)

Q1: Why does arcsin(1) return 90° but arccos(1) returns 0°?
A: This is by definition - arcsin gives the angle whose sine is the input, while arccos gives the angle whose cosine is the input.

Q2: What's the difference between radians and degrees?
A: Radians are the standard angular measure in mathematics (2π = 360°), while degrees are more common in everyday use.

Q3: Why can't I calculate arcsin(2)?
A: The sine function only outputs values between -1 and 1, so arcsin is only defined for inputs in that range.

Q4: How precise are the results?
A: Results are accurate to at least 6 decimal places, using PHP's built-in trigonometric functions.

Q5: Can I calculate inverse cosecant, secant, or cotangent?
A: This calculator focuses on the three primary inverse functions. Others can be derived from these (e.g., arccsc(x) = arcsin(1/x)).