Ever wondered how a calculator turns a sine value back into an angle? That’s the magic of inverse trigonometric functions.
In simple words, an inverse trig function takes a ratio like 0.5 and tells you which angle gives that ratio. Think of it as the reverse side of a seesaw: you know the height, you find the tilt.
What are Inverse Trigonometric Functions?
When you hear inverse (first time use: the opposite direction of something), you should think “undo”. The ordinary trig functions – sine, cosine, tangent – turn an angle into a ratio. Their inverses – arcsine, arccosine, arctangent (often written sin⁻¹, cos⁻¹, tan⁻¹) – do the opposite: they start with a ratio and give you an angle.
Why do we need them?
Imagine you measure the shadow of a pole and you know the height‑to‑shadow ratio. To find the sun’s elevation angle, you use an inverse trig function. In exams, you’ll often be given a ratio and asked for the angle, or you’ll need to solve equations like sinθ = 0.6.
Defining the Six Main Inverses
Each inverse function has a principal value (first time use: the standard answer we pick so the function is single‑valued). Because the original trig functions repeat every 360°, we restrict the output range to keep things tidy.
- arcsin x (sin⁻¹ x): gives an angle whose sine is x. Domain = [‑1, 1]; Range = [‑π/2, π/2] (‑90° to 90°).
- arccos x (cos⁻¹ x): gives an angle whose cosine is x. Domain = [‑1, 1]; Range = [0, π] (0° to 180°).
- arctan x (tan⁻¹ x): gives an angle whose tangent is x. Domain = all real numbers; Range = [‑π/2, π/2] (‑90° to 90°).
- arccot x (cot⁻¹ x): gives an angle whose cotangent is x. Domain = all real numbers; Range = (0, π) (0° to 180°).
- arcsec x (sec⁻¹ x): gives an angle whose secant (1/cosine) is x. Domain = (‑∞,‑1] ∪ [1,∞); Range = [0, π/2) ∪ (π/2, π].
- arccsc x (csc⁻¹ x): gives an angle whose cosecant (1/sine) is x. Domain = (‑∞,‑1] ∪ [1,∞); Range = [‑π/2, 0) ∪ (0, π/2].
Quick Comparison Table
| Function | Domain (allowed x) | Range (output θ) |
|---|---|---|
| arcsin x | ‑1 ≤ x ≤ 1 | ‑π/2 ≤ θ ≤ π/2 |
| arccos x | ‑1 ≤ x ≤ 1 | 0 ≤ θ ≤ π |
| arctan x | All real x | ‑π/2 ≤ θ ≤ π/2 |
| arccot x | All real x | 0 |
| arcsec x | |x| ≥ 1 | 0 ≤ θ |
| arccsc x | |x| ≥ 1 | ‑π/2 |
How to Use Inverse Trig Functions in Exams
Example 1 – Finding an Angle from a Sine Value
Find θ such that sin θ = ½ and 0 ≤ θ ≤ π/2.
Step 1: Recognise the inverse operation we need – arcsin.
Step 2: Apply it: θ = arcsin(½).
Using a calculator (or the unit‑circle memory), arcsin(½) = π/6 ≈ 30°.
Because the required range is the first‑quadrant (0 to π/2), this is the only answer.
Example 2 – Solving an Equation with tan⁻¹
Solve 2 tan⁻¹ x − π/4 = 0 for real x.
Step 1: Isolate the inverse: 2 tan⁻¹ x = π/4 → tan⁻¹ x = π/8.
Step 2: Apply the tangent (the opposite of tan⁻¹) to both sides: x = tan(π/8).
Using a calculator, tan(π/8) ≈ 0.4142. So x ≈ 0.414.
Tips to Remember
- Always check the domain first – if the given ratio lies outside the allowed interval, the expression is undefined.
- After finding the principal value, think about other possible angles using the periodic nature of trig functions (add 2πk, where k is any integer) if the question asks for “all solutions”.
- When the question involves compositions like sin⁻¹(cos θ), convert the inner function to a ratio first, then apply the inverse.
📝 Likely Exam Questions
- Find the principal value of cos⁻¹(‑0.3).
Answer: cos⁻¹(‑0.3) ≈ 1.875 rad (≈ 107.5°). - Solve sin⁻¹ x + cos⁻¹ x = π/2 for x ∈ [‑1,1].
Answer: Using the identity sin⁻¹ x + cos⁻¹ x = π/2, the equation holds for every x in the domain. So any x between ‑1 and 1 is a solution. - If tan θ = 3, find θ in the interval (‑π/2, π/2).
Answer: θ = tan⁻¹ 3 ≈ 1.249 rad (≈ 71.6°). - Evaluate sec⁻¹(2) and give its principal value in degrees.
Answer: sec⁻¹ 2 = arcsec 2 = π/3 ≈ 60° (since sec 60° = 2). - Solve for x: arcsin x + arcsin (2x) = π/2.
Answer: Using the identity arcsin a + arcsin b = π/2 ⇔ a² + b² = 1 and ab ≥ 0, we get x² + (2x)² = 1 → 5x² = 1 → x = ±√(1/5). Since both arcsin terms must be non‑negative, choose x = √(1/5) ≈ 0.447.