Inverse Trigonometric Functions Explained: arcsin, arccos, arctan
PUC Mathematics | Inverse Trigonometric Functions | Daniel Classes
Why Inverse Trig Functions Matter
If sin(π/6) = 1/2, then the inverse question is: if sin(x) = 1/2, what is x?
That’s where inverse trigonometric functions come in. They undo what trig functions do.
Core challenge: Trig functions are many-to-one (multiple angles give the same sine value). To have an inverse, we restrict the domain to make them one-to-one. That restriction defines the range of the inverse function.
Inverse Sine (arcsin or sin⁻¹)
Definition: If sin(θ) = x, then arcsin(x) = θ, where θ ∈ [-π/2, π/2].
Domain: [-1, 1]
Range: [-π/2, π/2]
Key values:
- arcsin(0) = 0
- arcsin(1/2) = π/6
- arcsin(1) = π/2
- arcsin(-1/2) = -π/6
Important: arcsin is defined only for values in [-1, 1]. arcsin(2) is undefined.
Inverse Cosine (arccos or cos⁻¹)
Definition: If cos(θ) = x, then arccos(x) = θ, where θ ∈ [0, π].
Domain: [-1, 1]
Range: [0, π]
Key values:
- arccos(1) = 0
- arccos(1/2) = π/3
- arccos(0) = π/2
- arccos(-1) = π
Note: The range is [0, π], not [-π/2, π/2]. This is the key difference from arcsin.
Inverse Tangent (arctan or tan⁻¹)
Definition: If tan(θ) = x, then arctan(x) = θ, where θ ∈ (-π/2, π/2).
Domain: All real numbers
Range: (-π/2, π/2)
Key values:
- arctan(0) = 0
- arctan(1) = π/4
- arctan(-1) = -π/4
- arctan(∞) approaches π/2
- arctan(-∞) approaches -π/2
Derivatives of Inverse Trig Functions
d/dx[arcsin(x)] = 1/√(1-x²)
d/dx[arccos(x)] = -1/√(1-x²)
d/dx[arctan(x)] = 1/(1+x²)
These are essential for calculus applications involving inverse trig functions.
How PUC Exams Test Inverse Trig Functions
Type 1: Evaluate Inverse Functions (2-3 marks)
Example: “Find arcsin(√3/2).”
Solution: We need θ ∈ [-π/2, π/2] such that sin(θ) = √3/2. That’s θ = π/3.
Type 2: Solve Equations with Inverses (3-5 marks)
Example: “Solve arcsin(x) = π/6.”
Solution: x = sin(π/6) = 1/2
Type 3: Simplify Expressions (3-5 marks)
Example: “Simplify sin(arccos(x)).”
Solution: Let θ = arccos(x), so cos(θ) = x and θ ∈ [0, π]. Then sin(θ) = √(1 – x²) [positive since θ ∈ [0, π] where sin is non-negative].
The Five Common Mistakes
Mistake 1: Confusing arcsin Range with arccos Range
Example: “Find arccos(1/2).”
Wrong: “arccos(1/2) = π/3, but I’ll also include -π/3.” [arcsin does this, not arccos]
Right: “arccos(1/2) = π/3 only. arccos has range [0, π], so negative angles are excluded.”
Fix: arcsin ∈ [-π/2, π/2]. arccos ∈ [0, π]. arctan ∈ (-π/2, π/2).
Mistake 2: Forgetting the Domain Restriction
Example: “Evaluate arcsin(2).”
Wrong: “The answer is 2.” [No angle has sine equal to 2]
Right: “Undefined. arcsin is only defined for x ∈ [-1, 1].”
Fix: Always check: is the input in the domain?
Mistake 3: Misinterpreting sin(arccos(x))
Example: “Simplify sin(arccos(x)).”
Wrong: “It equals 1 – x.” [Guessing]
Right: Let θ = arccos(x), so cos(θ) = x. Then sin²(θ) = 1 – cos²(θ) = 1 – x². Since θ ∈ [0, π], sin(θ) ≥ 0, so sin(θ) = √(1 – x²).
Fix: Use the Pythagorean identity and remember the range.
Mistake 4: Confusing arcsin(sin(x)) with Just x
Example: “Simplify arcsin(sin(2π/3)).”
Wrong: “It’s 2π/3.” [But 2π/3 is outside arcsin’s range]
Right: sin(2π/3) = √3/2. Since arcsin has range [-π/2, π/2], arcsin(√3/2) = π/3 ≠ 2π/3.
Fix: arcsin(sin(x)) = x only if x ∈ [-π/2, π/2]. Otherwise, reduce x into that range.
Mistake 5: Forgetting the Negative Sign in arccos Derivative
Example: “Find d/dx[arccos(x)].”
Wrong: “It’s 1/√(1-x²).” [Missing the negative]
Right: “It’s -1/√(1-x²).”
Fix: arcsin and arctan have positive derivatives. arccos has a negative derivative.
What to Do Next
To master evaluation: Evaluate 20 inverse trig expressions at key angles.
To master simplification: Simplify 10 expressions like sin(arccos(x)), cos(arcsin(x)), etc.
To master equations: Solve 10 equations involving inverse trig functions.
The Problem Vault Alternative
The Problem Vault has 65+ Inverse Trigonometric Functions problems.