2. Inverse Trigonometric Functions

6 topics2 h

Chapter 2 of NCERT Mathematics for Class 12: Inverse Trigonometric Functions. Two sections — Making a trigonometric function invertible and Computing with the inverse functions. Six videos, 2 h in all.

Worked answers to the 43 exercise questions in this chapter: Exercise 2.1 · Exercise 2.2 · Miscellaneous Exercise

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Part 1

Making a trigonometric function invertible

15 minWhy none of the six ratios has an inverse on its natural domainStudents are told the trigonometric ratios have no inverses, and most of them go looking for a value the sine never takes. There is none. Over 41 declared targets the sine covers every one - 0 unnamable, 0 refused, 0 astray - and so does each of the other five. The half that fails is the other half: 73 angles land on 13 values, and one output collects 7 of them.15 min · 12 parts17 minCutting the domain down until the function is one-one, and what a branch isThe six intervals the trigonometric ratios get cut down to look like six arbitrary conventions. They are not. Where every cut falls is forced - at a turning point or at a break, the only two places a function can stop being one-one - and across all six families, the number of cuts that are neither is 0. What IS a convention is which member of each infinite family gets promoted.17 min · 12 parts21 minThe six principal value branches, and the table that fixes themThe six intervals the trigonometric ratios get cut down to look like six arbitrary conventions. They are not. Where every cut falls is forced - at a turning point or at a break, the only two places a function can stop being one-one - and across all six families, the number of cuts that are neither is 0. What IS a convention is which member of each infinite family gets promoted.21 min · 12 parts19 minReading the graph of an inverse off the original by swapping the axes19 min · 12 parts

Part 2

Computing with the inverse functions

20 minWhen the two cancellation rules hold, and the angles where the second one failsTwo statements about inverse trigonometric functions sit one line apart and look like mirror images of each other. They are not. Over 361 angles spanning three turns either side of zero, the first rule holds for all six families with 0 failures, and the second holds for exactly the angles inside the promoted interval - 61 of the 361 for the sine, and 0 from outside it.20 min · 12 parts23 minSubstituting an angle for the variable to collapse a messy expression23 min · 14 parts

What you will be able to do

  • State the criterion the previous chapter fixed for a function to have an inverse, and apply it as a two-part test
  • State the two demands a surviving piece of the domain has to meet before an inverse can be built on it
  • Reproduce the six-row table from memory, with the correct bracket shape in every cell
  • State the coordinate fact that turns the graph of a function into the graph of its inverse
  • State both cancellation rules with the exact set of inputs each is valid on
  • Carry out the basic move: name the inverse value as an angle, so that the variable becomes a trigonometric ratio of it

What this chapter assumes you already know

  • The six trigonometric ratios as functions of a real variable, from Class XI
  • The graph of each ratio over several turns, and the values it repeats
  • Radian measure, and the standard angles as fractions of pi
  • Interval notation, including a set written as one set minus another
  • One-one and onto, and the fact from Chapter 1 that an inverse needs both
  • Composition of two functions, and what the identity function does
  • The graphs of all six trigonometric ratios over several turns, from Class XI
  • Where each ratio attains its largest and smallest values, and where it breaks

Where people usually slip up

Sentences students actually say, taken from the notes the videos were made from. Each one is answered on the page of the video it belongs to.

  • "The trigonometric functions have no inverse, full stop."
  • "They fail because they are not onto."
  • "Sine repeats every two pi, so it takes each value twice per turn — I can just halve the domain."
  • "A raised minus one always means one over the thing."
  • "The domain of the tangent is the real line."
  • "Secant and cosecant take every real value, like tangent does."
  • "The inverse of a function and its reflection are two different ideas."
  • "Once we restrict, we have changed the sine function."