Inverse Trigonometric Functions Calculator
Calculate arcsin(x), arccos(x), and arctan(x)
For arcsin and arccos: value must be between -1 and 1. For arctan: any real number.
How to Use This Calculator
Enter Input Value
Input the value (x) you want to find inverse trigonometric functions for. For arcsin and arccos, the value must be between -1 and 1. For arctan, any real number works.
Calculate
Click "Calculate All Inverse Functions" to see arcsin(x), arccos(x), and arctan(x) all at once.
Review Results
View all three inverse function results in both degrees and radians. Note the domain and range restrictions for each function.
Formula
Inverse Trigonometric Functions:
arcsin(x) = θ, where sin(θ) = x
Domain: [-1, 1] | Range: [-90°, 90°] or [-π/2, π/2]
arccos(x) = θ, where cos(θ) = x
Domain: [-1, 1] | Range: [0°, 180°] or [0, π]
arctan(x) = θ, where tan(θ) = x
Domain: (-∞, ∞) | Range: (-90°, 90°) or (-π/2, π/2)
Key Relationships:
- arcsin(x) + arccos(x) = 90° = π/2 (for x in [-1, 1])
- sin(arcsin(x)) = x, for x in [-1, 1]
- cos(arccos(x)) = x, for x in [-1, 1]
- tan(arctan(x)) = x, for all real x
About Inverse Trigonometric Functions Calculator
The Inverse Trigonometric Functions Calculator is a comprehensive tool that calculates all three main inverse trigonometric functions: arcsin (inverse sine), arccos (inverse cosine), and arctan (inverse tangent). These functions "undo" their respective trigonometric functions, finding the angle that produces a given trigonometric value.
What are Inverse Trigonometric Functions?
Inverse trigonometric functions find angles from trigonometric values. They are the inverse operations of sine, cosine, and tangent. For example, if sin(30°) = 0.5, then arcsin(0.5) = 30°. Each inverse function has specific domain and range restrictions to ensure they are functions (one input gives one output).
When to Use This Calculator
- Triangle Problems: Find angles when you know side ratios
- Vector Mathematics: Calculate angles between vectors
- Physics: Determine angles in force and motion problems
- Engineering: Solve problems involving angles in mechanical systems
- Comparison: Compare all three inverse functions for the same input
- Verification: Verify the relationship arcsin(x) + arccos(x) = 90°
Why Use Our Calculator?
- ✅ All Three Functions: Calculate arcsin, arccos, and arctan simultaneously
- ✅ Domain Validation: Clearly shows when functions are undefined
- ✅ Relationship Display: Shows that arcsin(x) + arccos(x) = 90°
- ✅ Multiple Units: Results in both degrees and radians
- ✅ 100% Free: No registration required
- ✅ Educational: Helps understand inverse trigonometric functions
Domain and Range Restrictions
- arcsin: Domain [-1, 1], Range [-90°, 90°] - returns angles in quadrants I and IV
- arccos: Domain [-1, 1], Range [0°, 180°] - returns angles in quadrants I and II
- arctan: Domain (-∞, ∞), Range (-90°, 90°) - returns angles in quadrants I and IV
Frequently Asked Questions
Why do arcsin and arccos only accept values between -1 and 1?
Because sine and cosine only produce values between -1 and 1 (as they represent coordinates on a unit circle). Since arcsin and arccos are their inverses, they can only "reverse" values in that range.
Why can arctan accept any real number?
Tangent can produce any real number as output (from -∞ to +∞), so its inverse arctan can accept any real number as input. The ratio of opposite/adjacent sides can be any value.
Why does arcsin(x) + arccos(x) = 90°?
For complementary angles, sin(θ) = cos(90° - θ). So if sin(θ) = x, then cos(90° - θ) = x, meaning arccos(x) = 90° - θ = 90° - arcsin(x). Rearranging gives arcsin(x) + arccos(x) = 90°.
What's the difference between sin⁻¹(x) and arcsin(x)?
They are the same function! Both notations represent the inverse sine function. arcsin is clearer because sin⁻¹ can be confused with 1/sin(x) (cosecant), which is the reciprocal, not the inverse.
Can inverse trigonometric functions return angles greater than 180°?
Only arccos can return angles up to 180° (its range is [0°, 180°]). Arcsin and arctan return angles only in the range [-90°, 90°]. These restrictions ensure each input has exactly one output (making them functions).