Inverse Trigonometric Functions â Class 12 ISC Mathematics revision notes
A 4-page visual revision sheet for Inverse Trigonometric Functions, written for Class 12 ISC Mathematics. Formulas, diagrams and the traps that cost marks, laid out to be read the night before rather than the term before. Free, and it opens inside CapCompass.
Page 1 as PDF
Inverse Trigonometric Functions in one look
six functions, six principal ranges, and the conditions on every identity
Why a restriction is needed at all
- sine takes the value 1/2 at infinitely many angles, so as it stands it is many to one and has no inverse
- the fix is to restrict the domain to an interval on which the function is one to one and still covers the whole range
- the interval chosen is called the principal branch, and the value it returns is the principal value
- so arc sin of 1/2 is Ī/6 and NOT the whole family nĪ + (-1)^n Ī/6, which belongs to solving equations rather than to this function
The six, with domain and principal range
| Function | Domain | Principal range |
|---|---|---|
| arc sin x | [-1, 1] | [-Ī/2, Ī/2] |
| arc cos x | [-1, 1] | [0, Ī] |
| arc tan x | all reals | (-Ī/2, Ī/2) |
| arc cot x | all reals | (0, Ī) |
| arc sec x | |x| >= 1 | [0, Ī] without Ī/2 |
| arc cosec x | |x| >= 1 | [-Ī/2, Ī/2] without 0 |
Reading a principal value off the range
- for a negative argument, arc sin, arc tan and arc cosec return a NEGATIVE angle, since their ranges straddle zero
- arc cos, arc cot and arc sec return an angle in the SECOND quadrant instead, because their ranges are 0 to Ī
- so arc sin(-1/2) is -Ī/6 while arc cos(-1/2) is 2Ī/3, and getting those two the same way round is the standard mistake
- the tidy way to state it: odd functions arc sin, arc tan and arc cosec satisfy f(-x) = -f(x); the other three satisfy f(-x) = Ī - f(x)
Page 1 of 4. The full sheet, all 4 pages, opens free in the app.
What this Inverse Trigonometric Functions sheet covers
- Why a restriction is needed at all
- The six, with domain and principal range
- Reading a principal value off the range
- The complementary and reciprocal pairs
- sine and cosine
- tangent and cotangent
- secant and cosecant
- reciprocal
- reciprocal, tangent
- The addition formulas, with their conditions
- tangent sum
- when xy > 1 and both positive
- when xy > 1 and both negative
- tangent difference
- double
- double, sine form
- double, cosine form
- Composing a function with an inverse
- The substitution that solves most problems
- Drawing the right triangle instead
- The graphs, in one look
- Solving an equation in inverse functions
- The standard results worth recognising instantly
- The telescoping series
- arc tan x + arc tan y: which case is xy in?
- Worked: composing an inverse with a double angle
- Traps that cost marks
- Tricks that save time
How CapCompass uses this sheet
These notes are not a standalone PDF. CapCompass plans Mathematics chapter by chapter against your real exam dates and free hours, and when the plan reaches Inverse Trigonometric Functions the sheet opens inside the study block itself â at the page covering the sub-topic that block is on, with the timer still running. Revision arrives at the moment the plan says revise, instead of having to be gone looking for.