The extreme cases at 0° and 90°, and the ratios that stop being defined

What to assume they know

• Defining sine, cosine and tangent, then their three reciprocals — the six ratios and which side sits underneath each one
• Squeezing 30°, 45° and 60° out of two special triangles — the exact values at 30°, 45° and 60°, which fill the middle of the same table
• That the two acute angles of a right triangle add to 90°, so one cannot reach 90° while the other survives
• That division by zero is not an operation, and that a reciprocal of zero does not exist
• Reading a value off a two-way table

What they should be able to do

• Explain why 0° and 90° cannot appear as the working angle in an actual right triangle
• Describe what happens to each side of the triangle as the angle is squeezed towards 0°, and again as it is opened towards 90°
• State the four defined values the chapter adopts at these two angles, and say on what grounds
• Derive the remaining values at 0° and at 90° from those definitions
• Identify which four entries of Table 8.1 are undefined and give the reason in each case
• Read Table 8.1 as a whole and describe how sine and cosine behave across the range
• Judge true-or-false claims about how these ratios change as the angle grows

Where it usually goes wrong

• "Not defined means the answer is infinity." It means the operation cannot be carried out. Nothing has a value here — there is no number, not a very large one.
• "Not defined means zero." The two are opposites in this table: an entry is undefined precisely where its reciprocal partner is zero.
• "0° and 90° are just two more angles." Neither can be the working angle of a right triangle. Everything in this topic is definition supported by a picture, not measurement.
• "The triangle in the last panel is very thin." In the last panel of each strip there is no triangle at all. The chapter draws the collapse, and that is the point.
• "The cosine increases with the angle, like the sine." It falls. This is the single most-tested confusion in the section, and Exercise 8.2 question 4 asks it directly.
• "sin θ = cos θ, since they're both about the same triangle." They agree at 45° and part company everywhere else. One row rises while the other falls; they can only cross once.
• "You can memorise the table as a block of symbols." The four gaps are the part that has to be reasoned, and they are also the part that is asked about. Anchor each gap to the side that vanished.

Questions to check understanding

• State the value of a named ratio at 0° or at 90°, or say that it is undefined
• Explain why a particular entry of the table is undefined
• True-or-false items on whether a named ratio grows or shrinks as the angle grows
• Evaluate an expression that mixes special-angle values with an endpoint value
• Identify the angle at which two named ratios are equal
• Complete a partially blanked copy of Table 8.1

Examples worth working on the board

Inputs only. Values marked verified are worked out here on the chapter's printed data.

• Fig. 8.16 (§8.3, p. 123). A single small right triangle ABC, A at the lower left, the right angle at B at the lower right, C above B. It is the starting frame for the shrinking argument.
• Fig. 8.17 (§8.3, p. 123) — a strip of six panels, which I counted on a close-up taken wide enough to include the figure caption. In every panel the base AB is the same length and B stays put; C slides down the vertical through B, so the angle at A closes. From the second panel onward a dotted outline of the first panel is laid over the current one — read, the dotted apex sits at one and the same height in all five later panels, and that height is panel 1's apex. So each frame carries exactly one ghost, the same one every time; it is not a cumulative fan and it is not the immediately preceding frame. The ghost is two dotted segments, the original hypotenuse and the stretch of the original vertical leg lying above where C has slid to. In the sixth and last panel C has arrived at B: the vertical leg has gone entirely and the figure is a single segment with the letter C printed just to the right of B.
• What the shrinking does to the ratios (§8.3, pp. 123–124). The vertical leg BC shrinks towards nothing while the hypotenuse AC settles onto the base AB. So the quotient BC/AC is heading for 0 and the quotient AB/AC is heading for 1. On that basis the chapter defines the sine of 0° as 0 and the cosine of 0° as 1.
• The rest of the 0° column, derived (§8.3, p. 124). Verified: the tangent is the sine over the cosine, which is 0 divided by 1, so 0. The secant is the reciprocal of the cosine, which is 1 divided by 1, so 1. The cotangent is the reciprocal of the tangent, and the tangent is 0, so there is nothing to invert. The cosecant is the reciprocal of the sine, and the sine is 0, so again there is nothing to invert. The chapter marks each of the two failures with a bracketed question; both answers are the same.
• Fig. 8.18 (§8.3, p. 124) — a strip of five panels, counted on a close-up taken wide enough to include the caption. Here it is A that moves: B and C stay fixed, with the right angle at B, and A slides right along the base towards B, so the angle at A opens out. The earlier positions of the base and hypotenuse are drawn dotted. In the fifth panel A has effectively reached B: the base has closed up to nothing and the hypotenuse lies along the vertical leg.
• The 90° definitions and the rest of the column (§8.3, p. 124). As the angle at A opens, the angle at C closes, the hypotenuse AC settles onto the vertical leg BC, and the base AB shrinks to nothing. So the sine is heading for 1 and the cosine for 0, and the chapter defines them as such. Verified for the remaining four: the cosecant is 1; the cotangent is the cosine over the sine, which is 0 over 1, so 0; the tangent would need a division by the cosine, which is 0, so it is undefined; and the secant would need the same division, so it is undefined too. The chapter hands these four to the reader rather than printing the working.
• Table 8.1 in full (§8.3, p. 125). Six rows against five columns; hand this to the explanation as data. Sine: 0, 1/2, 1/√2, √3/2, 1. Cosine: 1, √3/2, 1/√2, 1/2, 0. Tangent: 0, 1/√3, 1, √3, undefined. Cosecant: undefined, 2, √2, 2/√3, 1. Secant: 1, 2/√3, √2, 2, undefined. Cotangent: undefined, √3, 1, 1/√3, 0. The four undefined cells are the cosecant and cotangent at 0°, and the tangent and secant at 90°. Verified as a check on the whole table: in every column the cosecant is the sine inverted, the secant is the cosine inverted and the cotangent is the tangent inverted, and an undefined entry sits exactly where its partner is 0.
• The Remark (§8.3, p. 125). Across the range from 0° to 90° the sine climbs from 0 to 1 while the cosine falls from 1 to 0. This is stated for these two rows only; the chapter says nothing here about how the other four behave.
• The traps (Exercise 8.2 question 4, p. 127). Five true-or-false claims, of which three belong to this topic. Verified: the claim that the sine grows as the angle grows is true, and Table 8.1 is the evidence; the claim that the cosine grows as the angle grows is false, and the same row of the table refutes it; the claim that the cotangent is undefined at 0° is true. A fourth claim, that the sine and cosine agree for every angle, is false — the table shows them agreeing at 45° and nowhere else in the five columns.
• A value read straight from the endpoints (Exercise 8.2 question 2 item (iii), p. 127). The item asks for which listed angle a doubling relation between the sine of twice an angle and the sine of the angle holds. Verified: it holds at 0°, because both sides are 0. Of the three rival options, 30° and 45° are ruled out from Table 8.1 alone, since doubling them lands on 60° and on 90°, both of which the table supplies and neither of which agrees. The 60° option cannot be settled from this chapter at all — doubling it lands on 120°, and the chapter defines the ratios only for acute angles plus the two endpoints it adopts at 0° and 90°. The answer is still 0°, and the chapter's own data still establishes it; what the chapter's data does not do is eliminate the last rival.

Figures to have open

• The six panels of Fig. 8.17 redrawn as a schematic sequence, including the final collapsed frame and, on every panel after the first, the single dotted outline of the opening triangle — one ghost per panel, identical across the strip, not a spray of progressively shallower dotted hypotenuses. This is the chapter's own figure and the argument of sections 2 to 4 cannot be made without the progression.
• The five panels of Fig. 8.18 redrawn the same way, with the moving vertex clearly the one at the acute angle and the right angle staying put.
• Table 8.1 as a build-up graphic, with the four undefined cells highlighted and each linkable to the side that vanished. This is the chapter's own table; the explanation needs all thirty cells.
• A pair of small inset diagrams, one for each undefined pair, showing the zero denominator explicitly. Standard schematic.
• No photograph or textbook-only artwork is needed beyond the two strips and the table.

Where this sits in the book

• NCERT Mathematics, Textbook for Class X, Chapter 8 "Introduction to Trigonometry", §8.3, p. 123 — the opening of the 0° and 90° subsection, with Fig. 8.16 and Fig. 8.17
• §8.3, p. 124 — the 0° definitions and derivations, Fig. 8.18, and the 90° definitions
• §8.3, p. 125 — Table 8.1 and the Remark that follows it
• Exercise 8.2, p. 127, question 2 item (iii) and question 4 items (ii), (iii), (iv) and (v)
• The chapter summary, §8.5, p. 132, points 4 and 5 — the list of five angles, and the bounds that separate the two safe ratios from the rest
• Backward link inside this chapter: the bound on sine and cosine is first stated as a Remark in §8.2, p. 118