Why bother with inverse trigonometric functions?
Ever wondered how engineers turn angles back into numbers they can plug into a calculator? That's what inverse trig does – it tells you the angle when you know the ratio.
💡 In Simple Words: An inverse trigonometric function takes a value like 0.5 and tells you which angle has that sine, cosine or tangent. It’s like asking, “If the height of a triangle is half the hypotenuse, what’s the angle?”
What is an inverse trigonometric function?
An inverse trigonometric function (often called arcsine, arccosine or arctangent) is the opposite of a regular trig function. If sin θ = x, then the inverse sine, written sin⁻¹ x or arcsin x, gives you the angle θ. The same idea works for cosine and tangent.
Key point: we restrict the angle range
Because trig functions repeat every 360°, we pick a special interval so each value has only one angle. Think of it like a rulebook that says, “For this game, you can only play between 0° and 180°.”
Common inverse trig functions and their symbols
- sin⁻¹ x or arcsin x – inverse sine
- cos⁻¹ x or arccos x – inverse cosine
- tan⁻¹ x or arctan x – inverse tangent
Domain and range – the safe zones
Domain means the set of input numbers you’re allowed to plug in. Range means the set of angles you’ll get out. Here’s a quick snapshot:
| Function | Domain (input) | Range (output) |
|---|---|---|
| sin⁻¹ x | ‑1 ≤ x ≤ 1 | [-π/2 , π/2] (or –90° to 90°) |
| cos⁻¹ x | ‑1 ≤ x ≤ 1 | [0 , π] (or 0° to 180°) |
| tan⁻¹ x | All real numbers | (‑π/2 , π/2) (or –90° to 90°, not inclusive) |
How to solve a typical problem
Let’s walk through a classic ISC question.
Example 1: Find θ if sin θ = ½ and θ is in the first quadrant.
- Recognise the pattern: sin θ = ½ means θ could be 30° (π/6) or 150° (5π/6), but the first quadrant only allows angles up to 90°.
- Apply the inverse: θ = sin⁻¹(½). Using a calculator, sin⁻¹(0.5) = 30° (or π/6 rad).
- Check the quadrant: 30° is indeed in the first quadrant, so that’s the answer.
Result: θ = 30° (π/6 rad)
Example 2: Evaluate cos⁻¹(‑√3/2).
- Know that cosine of 150° (5π/6) equals –√3/2.
- Because the range of arccos is 0° to 180°, the angle we get is 150°.
Result: cos⁻¹(‑√3/2) = 150° (5π/6 rad)
Useful identities involving inverse functions
These shortcuts save time during exams.
- sin⁻¹ x + cos⁻¹ x = π/2 (90°) – the two angles always add up to a right angle.
- tan⁻¹ x + tan⁻¹ (1/x) = π/2 when x > 0.
- sin⁻¹ x = tan⁻¹( x / √(1‑x²) ) – handy if you only have a calculator for arctan.
Quick comparison of the three functions
- sin⁻¹: works only for inputs between –1 and 1; outputs angles between –90° and 90°.
- cos⁻¹: same input limits; outputs between 0° and 180°.
- tan⁻¹: accepts any real number; outputs between –90° and 90°, never actually reaching the endpoints.
Common pitfalls and how to avoid them
1. Mixing up degrees and radians. Always check the unit your calculator is set to.
2. Forgetting the range restriction. If you get an angle outside the allowed range, add or subtract 360° (or 2π) until it fits.
3. Using the wrong sign. Remember that sin⁻¹(‑0.5) gives –30°, not 150°.
📝 Likely Exam Questions
- Find the principal value of sin⁻¹(‑0.8).
Answer: sin⁻¹(‑0.8) ≈ –53.13° (or –0.927 rad). - If tan θ = 1, determine all possible values of θ in [0°, 360°].
Answer: θ = 45° or 225° (π/4 or 5π/4 rad). - Show that sin⁻¹ x + cos⁻¹ x = π/2 for any x in [‑1,1].
Answer: Let α = sin⁻¹ x, so sin α = x. Then cos α = √(1‑x²). Since cos⁻¹ x = π/2 – α, adding gives π/2. - Evaluate tan⁻¹(3) + tan⁻¹(1/3).
Answer: Using the identity, sum = π/2 (90°). - Solve for θ: cos⁻¹(2/3) = θ.
Answer: θ ≈ 48.19° (or 0.841 rad).