Reading the graph of an inverse off the original by swapping the axes

What to assume they know

• The graphs of all six trigonometric ratios over several turns, from Class XI
• Plotting a point from its two coordinates, and reading a point off a plot
• The line whose height always equals its horizontal position, and reflection in it
• The six promoted output intervals from the previous topic
• Which inputs each ratio is undefined at, and what its graph does there

What they should be able to do

• State the coordinate fact that turns the graph of a function into the graph of its inverse
• Carry out the swap on a named point and say where the image lands
• State the equivalent geometric operation and name the line it is taken about
• Identify, on any of the chapter's twelve inverse-function panels, which part of the drawn curve is the promoted branch
• Read the stroke-weight convention correctly, and say what the thin portions are
• Explain why the drawn curve for the inverse cosecant and the inverse secant shows the promoted branch as two separate arcs
• Explain why nothing is drawn in the strip between minus one and one on those two panels, and why nothing is drawn outside it on two others
• Say what the long dashed horizontal lines mark on the four unbounded panels
• Read a principal value off a graph, and say why the graph illustrates rather than proves it

Where it usually goes wrong

• "The thin parts of the curve are not part of the inverse function." They are parts of other branches, each a genuine inverse on its own interval. The heavy stroke marks the one the chapter promotes; it does not mark the only real one. The panels would be dishonest if they showed the heavy part alone, which is presumably why they do not.
• "The thin parts are drawn dashed." They are not, on any of the twelve panels. They are the same colour and thinner. The dashes on these pages are either short leader ticks from an axis label or, on the four unbounded panels, long horizontal levels — and neither of those is the curve.
• "The dashed horizontal lines are where the branches end." They are the heights the branches run towards without arriving. Branch boundaries on these panels are not drawn as lines at all; they are read off the heavy stroke's two ends.
• "Reflecting in the diagonal is a different construction from swapping the coordinates." They are the same construction. The chapter gives the coordinate version first and the reflection version second, and offers one panel where the two can be seen to agree.
• "The graph proves the inverse exists." It does not. The restriction settled that, in the previous two topics, by an argument about values. The picture is how you remember the result, and it is also where a wrong restriction becomes visible — which is a real use, but not a proof.
• "The inverse secant panel is broken into two arcs because the function is discontinuous there." The two arcs sit on either side of an empty vertical strip, and they are two because the promoted interval of outputs has a point removed from its middle. Two different holes, in two different directions, and students routinely merge them.
• "The inverse sine curve carries on past one." It stops. Nothing is drawn beyond minus one and one on the inverse sine and inverse cosine panels, because nothing outside that strip is accepted.
• "Axis labels on an inverse panel mean the same as on the original panel." They do not. On an original panel the horizontal axis carries angles; on an inverse panel the vertical axis does. Every panel pair on Part I pp. 20 to 25 makes this switch, and it is the single fact a student must hold to read them.

Questions to check understanding

• Given one plotted point of a trigonometric function, write down the partner point that lies on its inverse
• State the line about which the two graphs are mirror images, and use it to locate one point from another
• Given a drawing of an inverse function with several branches shown, identify the promoted one and give its two endpoints
• Explain why the drawn curve for a named inverse function has a gap in the middle, or stops at minus one and one
• Say what a stated dashed horizontal level on one of these panels represents
• Read a principal value off a supplied panel, and give one non-principal angle with the same trigonometric ratio
• Sketch the promoted branch of a named inverse function with correct endpoints and correct open or closed dots

Examples worth working on the board

Values marked verified are worked out here from the chapter's own printed data; neither answers file was opened, and this chapter prints no answers to its exercises.

• Remark (i) (§2.2, Part I pp. 19–20). The chapter states the coordinate fact first: for an invertible function, a point on the original graph has a partner on the inverse graph with the two coordinates exchanged. It then draws the conclusion about the whole picture. Note the order — the point comes first and the picture second.
• A worked swap (not in the book, on the chapter's own sine). Verified: the sine of pi by six is one half, so the point with horizontal position pi by six and height one half lies on the sine graph, and the point with horizontal position one half and height pi by six lies on the inverse sine graph. Read the second point off Fig 2.1 (ii) on the page and it is there. Do this once, slowly, before any wholesale redrawing.
• Remark (ii) (§2.2, Part I p. 20). The same operation described as a reflection about the line whose height always equals its horizontal position, with Fig 2.1 (iii) offered as the way to see it.
• Fig 2.1, three panels (Part I p. 20). Panel (i) is the sine over roughly five half-turns either side of the origin, angle on the horizontal axis, marked at the multiples of half pi, with one and minus one marked on the vertical. Panel (ii) is the inverse sine, standing up: the angle is now on the vertical axis, marked from minus five pi by two to five pi by two, and only minus one and one are marked on the horizontal. Panel (iii) carries both curves on one pair of axes together with a straight arrowed line through the origin, labelled with its own equation. Read off the printed page and confirmed on the printed page: in panel (ii) the segment running between minus half pi and half pi is drawn at a visibly heavier stroke than the rest of the curve, and the rest of the curve is not dashed — it is the same colour and simply thinner. The only dashed marks in that panel are short leader ticks from the axis labels to the curve.
• The chapter's caption sentence (Part I p. 20). It says that the heavily drawn part of the inverse sine panel is the promoted branch. That one sentence is the key to all twelve inverse panels, and it is printed only once, under the first of them.
• Fig 2.2, two panels (Part I p. 21). The cosine, and the inverse cosine standing up. Verified on the printed page: the heavy portion of panel (ii) runs from the point at horizontal position one and height zero, up to the point at horizontal position minus one and height pi — exactly the promoted interval — and the curve is drawn nowhere outside the strip from minus one to one.
• Fig 2.3, two panels (Part I p. 22), and Fig 2.4, two panels (Part I p. 23). The cosecant and the secant, and their inverses. Verified on the printed page of both: on each inverse panel the heavy stroke appears as two separate arcs, one to the left of minus one and one to the right of one, because the promoted branch is an interval with an interior point missing and that missing point is exactly where the two arcs would have joined. Also verified: no curve at all is drawn in the strip strictly between minus one and one on either panel. Long dashed horizontal lines run right across both panels — at the whole multiples of pi on the inverse cosecant panel, and at the odd multiples of half pi on the inverse secant panel.
• Fig 2.5, two panels (Part I p. 24), and Fig 2.6, two panels (Part I p. 25). The tangent and the cotangent, and their inverses. Verified on Fig 2.5 (ii): the promoted branch is a single heavy S-shaped curve through the origin, three thinner copies of it sit above and below, and three long dashed horizontal lines are drawn, at half pi, at three half pi and at minus half pi. The corresponding dashed lines on the inverse cotangent panel sit at the whole multiples of pi.
• What the dashed lines mark (not in the book). Verified against the four panels: each dashed level is a height the branches approach as the horizontal position runs away to either side, and never reach. For the inverse tangent and the inverse secant those levels are the odd multiples of half pi; for the inverse cotangent and the inverse cosecant they are the whole multiples of pi. They are not branch boundaries and they are not gridlines. Say so explicitly, because a student who reads them as boundaries will put the inverse tangent's promoted branch in the wrong place.
• Reading a principal value off a panel (not in the book, on Fig 2.1 (ii)). Verified: stand at horizontal position one half on the inverse sine panel, go up until the heavy stroke is met, and read the height as pi by six. Stand at the same place and carry on to the next thin crossing and the height is five pi by six, which is a perfectly good angle whose sine is one half and is not the principal value. The two crossings on one vertical line are the clearest statement in the chapter of what the promotion is for.
• The delegation on Part I p. 21. Rather than repeat the argument, the chapter says the inverse cosine panel is drawn the same way as the inverse sine panel was. Build the method once on the sine, then move quickly.

Figures to have open

• Redraws of the chapter's own panels, which are the substance of this topic: Fig 2.1 (i), (ii) and (iii) from Part I p. 20; Fig 2.2 (i) and (ii) from Part I p. 21; Fig 2.3 (i) and (ii) from Part I p. 22; Fig 2.4 (i) and (ii) from Part I p. 23; Fig 2.5 (i) and (ii) from Part I p. 24; Fig 2.6 (i) and (ii) from Part I p. 25. Thirteen panels in all; only the first figure has a third panel.
• The redraws must preserve three things exactly: the stroke-weight difference between the promoted branch and the rest; the empty vertical strip on the inverse cosecant and inverse secant panels; and the position of the long dashed levels, which differ between the two pairs. Everything else — colour, axis styling, label placement — may be restyled.
• A legend card for section 6 distinguishing heavy stroke, thin stroke, leader tick and dashed level. The chapter has no legend and needs one.
• One clean diagonal-reflection construction for section 4, with a point, its mirror, the joining segment and the two equal halves marked. Not in the book; the chapter asserts the reflection and draws no construction.
• The peers rule applies with force here: the twelve inverse panels are siblings and must be laid out at one size and one scale, or the reader will read importance into a difference that is only layout.

Where this sits in the book

• NCERT Class 12 Mathematics, Chapter 2 "Inverse Trigonometric Functions", §2.2 Basic Concepts, Remarks (i) and (ii), Part I pp. 19–20
• Fig 2.1 (i), (ii) and (iii) with the caption sentence about the heavily drawn part, Part I p. 20
• Fig 2.2 (i) and (ii) and the sentence delegating the construction, Part I p. 21
• Fig 2.3 (i) and (ii), Part I p. 22; Fig 2.4 (i) and (ii), Part I p. 23
• Fig 2.5 (i) and (ii), Part I p. 24; Fig 2.6 (i) and (ii), Part I p. 25