Inverse Trigonometric Functions
Class 12 · Mathematics
4 ideas in Class 12 Mathematics, explained in plain words, with games and quizzes to practise.
Principal Branches and Domains
Part of: Branches and Principal Values
Sine, cosine and friends repeat forever, so no single undo exists until the input is fenced into one stretch where the function climbs or falls cleanly. Each fenced stretch is a branch, and the agreed favourite is the principal branch. Its input-output table, like [−1, 1] to [−π/2, π/2] for sin⁻¹, is the principal domain and range.
Why it matters. Every later value and identity lives inside these fences, and mixing branches is the top error source.
Formulas
- sin⁻¹ takes [−1, 1] to [−π/2, π/2]
- cos⁻¹ takes [−1, 1] to [0, π]
- tan⁻¹ takes all reals to (−π/2, π/2)
Key words
- Principal value branch
- : The agreed favourite input stretch used for an inverse trig function.
- Principal value
- : The output lying inside the principal branch.
- Branch of a function
- : One clean stretch of a repeating function that can be inverted.
- One-one restriction
- : Fencing a function until distinct inputs give distinct outputs.
- Mirror property
- : An inverse graph mirrors the original in the line y = x.
Try it in two minutes
- Sketch one sine wave and fence a single clean climb.
- Read inputs and outputs only inside the fence.
- Repeat the fencing for cosine on [0, π].
Try at home. Write the domain and principal range of cot⁻¹ from memory and check them.
Principal Values of Standard Inputs
Part of: Branches and Principal Values
A principal value asks which fenced angle produces the given input. Sine and tangent inverses keep the sign of the input, while cosine and secant inverses bend negatives into the second quadrant. Secant, cosecant and cotangent values reduce to their sine, cosine and tangent cousins.
Why it matters. Fast exact values power every identity and equation that follows.
Formulas
- Negative inputs give negative outputs for sin⁻¹ and tan⁻¹
- cos⁻¹(1/2) is π/3; sec⁻¹(−2) is 2π/3
- sec⁻¹x equals cos⁻¹(1 over x)
Key words
- Sine inverse range
- : The fence [−π/2, π/2] holding every sin⁻¹ value.
- Cosine inverse range
- : The fence [0, π] holding every cos⁻¹ value.
- Tangent inverse range
- : The open fence (−π/2, π/2) holding every tan⁻¹ value.
- Secant inverse domain
- : All reals except the open interval (−1, 1).
- Cotangent inverse range
- : The open fence (0, π) holding every cot⁻¹ value.
- Cosecant inverse domain
- : All reals except the open interval (−1, 1), with zero removed from range.
Try it in two minutes
- Ask which fenced angle has the given sine.
- Check the sign rule for the function family.
- Reduce sec⁻¹, cosec⁻¹ and cot⁻¹ to the big three.
Try at home. Find tan⁻¹(−√3) and cos⁻¹(−1/2) from scratch and check the fences.
Cancelling Inside and Outside the Branch
Part of: Undoing and Simplifying
Applying a function after its inverse cancels cleanly, but only where each is defined: sin(sin⁻¹x) is x on [−1, 1], while sin⁻¹(sin x) is x only inside [−π/2, π/2]. Outside angles first reduce by periodicity and symmetry into the fence, and complementary pairs always sum to π/2.
Why it matters. Nested expressions are the standard trap: cancelling blindly outside the fence gives wrong angles.
Formulas
- sin(sin⁻¹x) is x on [−1, 1]
- sin⁻¹(sin x) is x only inside [−π/2, π/2]
- sin⁻¹x plus cos⁻¹x is π/2
Key words
- Inner-outer cancellation
- : A function undoing its own inverse on matching domains.
- Principal interval
- : The fenced angle stretch where direct cancellation is valid.
- Complementary identity
- : The rule that sin⁻¹x plus cos⁻¹x is π/2.
- Reduction to branch
- : Sliding an outside angle into the fence using symmetry.
Try it in two minutes
- Check whether the inner value sits inside the fence.
- If not, reduce it with symmetry into the fence.
- Read the answer off the reduced angle.
Try at home. Evaluate cos⁻¹(cos 5π/4) by reducing into [0, π] first.
Identities and Simplest Forms
Part of: Undoing and Simplifying
Sums of inverses follow angle-addition rules only when the inputs stay small; large inputs push the sum into the next quadrant. Messy expressions simplify by writing the variable as a trig ratio of a new angle, reading off the answer, and checking the fence at the end.
Why it matters. Simplest-form questions carry heavy marks and punish blind formula use.
Formulas
- tan⁻¹x plus tan⁻¹y needs care when xy crosses 1
- Simplest forms come from writing x as a trig ratio
- Check the branch before cancelling
Key words
- Addition formula
- : The rule for combining two inverse tangents with a size condition.
- Simplest form
- : An expression rewritten as a single plain inverse or angle.
- Substitution method
- : Writing x as a trig ratio of a new angle to simplify.
- Validity interval
- : The input range where an identity truly holds.
Try it in two minutes
- Test the size condition before adding two inverses.
- Substitute x as a sine, cosine or tangent of a new angle.
- Simplify, then verify the result sits in the fence.
Try at home. Show tan⁻¹(1/2) + tan⁻¹(1/3) equals π/4 using the size condition.
What your child will practise
- Cancelling Inside and Outside the Branch: Candy Puzzle (Clue Puzzle)
- Cancelling Inside and Outside the Branch: Ocean Quick Fire (Quick Fire)
- Identities and Simplest Forms: Desert Puzzle (Clue Puzzle)
- Principal Branches and Domains: Candy Quick Fire (Quick Fire)
- Principal Values of Standard Inputs: Candy Quest (Flip and Match)
- Quiz: Cancelling Inside and Outside the Branch (8 questions)
- Quiz: Identities and Simplest Forms (8 questions)
- Quiz: Principal Branches and Domains (8 questions)
- Quiz: Principal Values of Standard Inputs (8 questions)