Inverse trig result
Principal angle
Enter a ratio
Degrees and radians will appear here.
What are Inverse Trigonometric Functions?
Inverse trigonometric functions (also called arcus functions or anti-trigonometric functions) are the inverse functions of the basic trigonometric functions. They are used to find the angle when given a trigonometric ratio.
For example, if sin(θ) = 0.5, then θ = arcsin(0.5) = 30°. The prefix "arc" indicates that we're finding an arc (angle) on the unit circle.
How to Calculate
- Identify the Input Range: arcsin and arccos require inputs between -1 and 1. arctan accepts any real number.
- Apply the Function: Use a calculator or tables to find the angle. Results are typically in the principal value range.
- Convert Units: Most calculators give results in radians by default. Multiply by 180/π to convert to degrees.
Formulas
arcsin(x) = sin⁻¹(x), where -1 ≤ x ≤ 1, range: [-π/2, π/2]
arccos(x) = cos⁻¹(x), where -1 ≤ x ≤ 1, range: [0, π]
arctan(x) = tan⁻¹(x), where x ∈ ℝ, range: (-π/2, π/2)
Examples
Calculate arcsin(0.5)
Problem: Find the angle whose sine is 0.5.
Solution:
- arcsin(0.5) = 30° or π/6 radians
- Result: 30° (or 0.5236 radians)
Calculate arccos(0.5)
Problem: Find the angle whose cosine is 0.5.
Solution:
- arccos(0.5) = 60° or π/3 radians
- Result: 60° (or 1.0472 radians)
Quick Calculation Tips
- arcsin and arccos only accept values between -1 and 1
- arctan can accept any real number
- Common values: arcsin(1) = 90°, arccos(0) = 90°, arctan(1) = 45°
- Results are principal values - there may be other valid angles
Common Mistakes to Avoid
-
Input out of range
arcsin(2) is undefined - sine values can only be between -1 and 1.
-
Confusing with reciprocals
arcsin(x) ≠ 1/sin(x). The "arc" prefix means inverse function, not reciprocal.
Frequently Asked Questions
What is the difference between sin⁻¹ and 1/sin?
sin⁻¹(x) (or arcsin) is the inverse function that finds the angle. 1/sin(x) is the reciprocal (cosecant). They are completely different operations.
Why are there multiple angles with the same sine?
sin(30°) = sin(150°) = 0.5. The principal value of arcsin(0.5) is 30°. You may need to consider the quadrant to find other valid angles.
What is arctan(∞)?
As x approaches infinity, arctan(x) approaches 90° (π/2 radians). This is why tan(90°) is undefined - it's essentially infinity.