Arcsine, Arccosine, Arctangent Calculator

The inverse trigonometric functions, commonly referred to as arcsine, arccosine, and arctangent, are used to determine the angles that correspond to given trigonometric values. These functions are the inverse operations of sine, cosine, and tangent, respectively.

Formulas

• θ = arcsin(x) - This formula gives the angle whose sine is x.
• θ = arccos(x) - This formula gives the angle whose cosine is x.
• θ = arctan(x) - This formula gives the angle whose tangent is x.

Examples

Example 1: For x = 0.5

• arcsin(0.5) = 30°
• arccos(0.5) = 60°
• arctan(0.5) ≈ 26.57°

Example 2: For x = 1

• arcsin(1) = 90°
• arccos(1) = 0°
• arctan(1) = 45°

Table of Common Values

x Value	Arcsine (arcsin(x))	Arccosine (arccos(x))	Arctangent (arctan(x))
0	0°	90°	0°
0.5	30°	60°	≈26.57°
1	90°	0°	45°
√3/2	60°	30°	≈56.31°

FAQs

What do arcsine, arccosine, and arctangent represent?

They are inverse trigonometric functions. While the regular trigonometric functions (sine, cosine, tangent) give the ratios of sides in a right triangle for a given angle, their inverse counterparts (arcsine, arccosine, arctangent) return the angle for a given ratio.

When would I use the arctangent function over arcsine or arccosine?

Arctangent is often used in situations involving slopes and tangents, especially in fields like physics and engineering. If you know the ratio of the opposite side to the adjacent side (the slope), arctangent will give you the angle. Arcsine and arccosine are more commonly used when you have the ratio of the opposite side to the hypotenuse or the adjacent side to the hypotenuse, respectively.

Why are the domains for arcsine and arccosine restricted to [-1,1]?

The sine and cosine functions yield outputs between -1 and 1, inclusive. As arcsine and arccosine are their inverses, they can only accept values in this range to return real number angles.

What's the difference between arctan and atan?

There's no mathematical difference; they represent the same function. "atan" is simply a shorter notation used by some mathematicians and most programming languages.

How do inverse trigonometric functions relate to complex numbers?

While the real-number outputs of arcsine and arccosine are restricted due to their domain, these functions can return complex number results when given inputs outside the [-1,1] range. This extends the functionality of the inverse trig functions into the complex plane.