Inverse Trigonometric Functions

Regular trigonometric functions take an angle and give a ratio. Inverse trigonometric functions do the opposite -- they take a ratio and give an angle. For example, if sin 30 = 0.5, then sin⁻¹(0.5) = 30 degrees.

Definition

Since trigonometric functions are not one-to-one over their entire domain, we restrict their domains to make them invertible:

| Function | Domain Restriction | Range (Principal Value) | |----------|-------------------|----------------------| | sin⁻¹(x) or arcsin(x) | [-1, 1] | [-pi/2, pi/2] | | cos⁻¹(x) or arccos(x) | [-1, 1] | [0, pi] | | tan⁻¹(x) or arctan(x) | (-inf, inf) | (-pi/2, pi/2) |

Important: sin⁻¹(x) does NOT mean 1/sin(x). It means "the angle whose sine is x."

Key Properties

sin⁻¹(-x) = -sin⁻¹(x) (odd function)

cos⁻¹(-x) = pi - cos⁻¹(x)

tan⁻¹(-x) = -tan⁻¹(x) (odd function)

sin⁻¹(x) + cos⁻¹(x) = pi/2 for all x in [-1, 1]

tan⁻¹(x) + cot⁻¹(x) = pi/2

Worked Examples

Example 1: Find sin⁻¹(sqrt(3)/2).

Solution: We need the angle theta in [-pi/2, pi/2] such that sin(theta) = sqrt(3)/2.

• sin(60) = sin(pi/3) = sqrt(3)/2
• sin⁻¹(sqrt(3)/2) = pi/3 (or 60 degrees)

Example 2: Find the value of sin(cos⁻¹(3/5)).

Solution:

• Let theta = cos⁻¹(3/5), so cos(theta) = 3/5
• Using sin²(theta) + cos²(theta) = 1: sin(theta) = 4/5 (positive since theta is in [0, pi])
• sin(cos⁻¹(3/5)) = 4/5

Example 3: Simplify tan⁻¹(1) + tan⁻¹(2) + tan⁻¹(3).

We can use the formula: tan⁻¹(a) + tan⁻¹(b) = tan⁻¹((a+b)/(1-ab)) + pi (when ab > 1)

• tan⁻¹(1) = pi/4
• tan⁻¹(2) + tan⁻¹(3) = pi + tan⁻¹((2+3)/(1-6)) = pi + tan⁻¹(-1) = pi - pi/4 = 3pi/4
• Total = pi/4 + 3pi/4 = pi

Graphs

• sin⁻¹(x): Increases from -pi/2 to pi/2 as x goes from -1 to 1
• cos⁻¹(x): Decreases from pi to 0 as x goes from -1 to 1
• tan⁻¹(x): Increases from -pi/2 to pi/2, with horizontal asymptotes

Simple Equations

To solve sin⁻¹(x) = pi/6:

• x = sin(pi/6) = 1/2

To solve cos⁻¹(2x) = pi/3:

• 2x = cos(pi/3) = 1/2
• x = 1/4

Key Takeaways

• Inverse trig functions return angles from given ratios
• Principal value ranges ensure unique outputs
• sin⁻¹(x) + cos⁻¹(x) = pi/2 is a frequently used identity
• Always check that the input is within the valid domain