Which of the six stays positive where, and what happens at the quarter turns

What to assume they know

• Coordinates on the unit circle extend the ratios to every real number — cosine and sine read off as the two coordinates where the terminal side cuts the unit circle, and the four quotient functions with their exclusions
• Signs of coordinates in the four quadrants of the plane
• That a non-zero number and its reciprocal share a sign, and that a quotient's sign is the product of the two signs
• Solving a squared value for its two roots and choosing between them

What they should be able to do

• Derive the sign of any of the six functions in any quadrant from the coordinates alone
• Explain why the cosecant row of the sign table copies the sine row and the secant row copies the cosine row
• Explain why the tangent and cotangent rows are identical to each other
• Derive the effect of negating the input on the cosine and on the sine, using the reflection of Fig 3.7
• State the bounds on the sine and cosine of any real number and say where they come from
• Explain why the quadrant descriptions use strict inequalities and what is true at the excluded points
• Given one function's value and the quadrant, find the remaining five with correct signs

Where it usually goes wrong

• "Twenty-four entries to memorise." Eight, and really two. If a student cannot rebuild the table from the coordinate signs in under half a minute, they have learnt the wrong thing.
• "The cosecant is the opposite of the sine, so its sign flips." A reciprocal never changes sign. One over a negative number is negative. This is the single most common error in the table.
• "Negating the input negates both functions." Only the sine. The reflection of Fig 3.7 moves the point vertically and leaves the first coordinate untouched.
• "Everything is negative in the second quadrant except the sine." The cosecant is positive there too, because it copies the sine.
• "At a quarter turn the cosine is positive because it is not negative." It is zero, which is neither, and the secant and tangent do not exist there at all. The strict inequalities on p. 52 are the chapter saying so.
• "Given the sine you can find the cosine." Only up to sign. Every one of Exercise 3.2's first five questions states a quadrant, and the statement is not decoration — remove it and each question has two answers.
• "A quadrant tells you the sign of everything, so no working is needed." The quadrant fixes the signs; the identities still have to supply the sizes.

Questions to check understanding

• State the sign of a named function in a named quadrant, with a reason
• Given the value of one function and the quadrant, find the other five
• Decide which quadrant a point is in from the signs of two named functions
• Explain why negating the input leaves the cosine alone
• Say what is true of each of the six functions at a quadrantal value
• Spot the error in a worked solution that has chosen the wrong root

Examples worth working on the board

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

• Fig 3.7 (§3.3.1, p. 51). The same unit circle as Fig 3.6, with A, B, C and D again lettered and their coordinate pairs printed. A point P carrying the pair (a, b) sits in the first quadrant with the arc from A to P marked x; directly below it, in the fourth quadrant, sits Q carrying the pair (a, −b), with the arc from A down to Q marked −x. The two arcs and both coordinate pairs are lettered inside the artwork.
• What the reflection gives (§3.3.1, p. 51). Reflecting in the horizontal axis leaves the first coordinate alone and reverses the second. Verified: therefore negating the input leaves the cosine unchanged and reverses the sign of the sine. These two statements reappear as the first two numbered results of §3.4 on p. 57, where the whole identity chain is built on them — so this small picture is doing more work than its position on the page suggests.
• The bounds (§3.3.1, pp. 51–52). Every point of the unit circle has both coordinates between −1 and 1 inclusive. Verified: hence the sine and cosine of any real number lie in that closed interval, and any working that produces 1.2 for either is wrong before it is checked against anything else.
• The quadrant descriptions (§3.3.1, p. 52). The page describes the four quadrants by strict inequalities on the arc: between 0 and a quarter turn, between a quarter and a half, between a half and three quarters, between three quarters and a full turn. Verified: the four open intervals together miss exactly the five quadrantal values in that range, which is the content of section 9.
• The sign table (§3.3.1, p. 52). Six rows and four columns. Reading across quadrants one to four:

| Function | I | II | III | IV | |---|---|---|---|---| | sine | + | + | − | − | | cosine | + | − | − | + | | tangent | + | − | + | − | | cosecant | + | + | − | − | | secant | + | − | − | + | | cotangent | + | − | + | − |

Verified as structure: the fourth row duplicates the first and the fifth duplicates the second, because a reciprocal keeps its sign; the third and sixth are identical to each other and are the products of the first two, because both are quotients of the same pair. Eight cells generate all twenty-four.

• Example 6 (§3.3, p. 55). Given cosine equal to −3/5 with the point in the third quadrant. Inputs only. Verified: secant −5/3; the first identity gives a squared sine of 16/25, so the sine is ±4/5 and the quadrant selects −4/5; cosecant −5/4; tangent 4/3; cotangent 3/4. Note the tangent comes out positive from two negatives, exactly as the table's third row says.
• Example 7 (§3.3, p. 56). Given cotangent equal to −5/12 with the point in the second quadrant. Inputs only. Verified: tangent −12/5; one plus the squared tangent gives a squared secant of 169/25, so the secant is ±13/5 and the quadrant selects −13/5; cosine −5/13; sine 12/13; cosecant 13/12. The route here goes through the second identity rather than the first, which is the reason the chapter prints two examples instead of one.
• Exercise 3.2, questions 1 to 5 (p. 57). Hand the data over intact. Q1 cosine −1/2, third quadrant. Q2 sine 3/5, second quadrant. Q3 cotangent 3/4, third quadrant. Q4 secant 13/5, fourth quadrant. Q5 tangent −5/12, second quadrant. Verified, as working added here: Q1 gives sine −√3/2, tangent √3, cosecant −2/√3, secant −2, cotangent 1/√3. Q2 gives cosine −4/5, tangent −3/4, cosecant 5/3, secant −5/4, cotangent −4/3. Q3 gives tangent 4/3, sine −4/5, cosine −3/5, cosecant −5/4, secant −5/3. Q4 gives cosine 5/13, sine −12/13, tangent −12/5, cosecant −13/12, cotangent −5/12. Q5 gives cosine −12/13, sine 5/13, secant −13/12, cosecant 13/5, cotangent −12/5. Note the secant and the cosecant here: the secant is the reciprocal of the cosine and the cosecant the reciprocal of the sine, so the two denominators are 12 and 5 respectively and they must not be interchanged.
• What is actually true on a boundary (an added tally, from the coordinate pairs printed in Fig 3.6 and reused in Fig 3.7). Verified at all four quadrantal points, and the pattern is the same at each: one coordinate is 0 and the other is ±1, so exactly two of the six functions take the value 0, exactly two fail to exist, and exactly two take the value 1 or −1. At the start of the turn, for instance, the sine and the tangent are 0, the cotangent and the cosecant do not exist, and the cosine and the secant are both 1. This is why the boundaries cannot be given a column in the sign table: they are not a mixture of pluses and minuses, they are a different kind of entry altogether. Note that every one of the five hands the student a value whose sign is already consistent with two different quadrants, so the stated quadrant is doing real work throughout the set — that is what the sign table is for. What singles out Q3 and Q5 is something else: they are the two given as a cotangent and a tangent, so the route to the remaining values runs through the second identity rather than the first.

Figures to have open

• Fig 3.7 redrawn with P, its mirror image Q, both coordinate pairs and both arcs marked. The chapter's own figure, and sections 2 and 3 depend on it. The lettering is inside the printed artwork.
• The unit circle with the four quadrants shaded and the four axis directions drawn as gaps, so the boundary exclusions are visible rather than stated. An added construction.
• The six-by-four sign table, built row by row rather than displayed complete. This is the chapter's own table on p. 52.
• A pair of worked-solution panels for Examples 6 and 7, side by side, so the two different identity routes can be compared. The chapter prints no figure for either.

Where this sits in the book

• NCERT Mathematics, Textbook for Class XI, Chapter 3 "Trigonometric Functions", §3.3.1 Sign of trigonometric functions, pp. 51–52 — Fig 3.7, the two reflection identities, the bounds, the quadrant descriptions and the sign table
• §3.3, p. 55 — Example 6; p. 56 — Example 7
• Exercise 3.2, questions 1 to 5, p. 57
• The same two reflection identities are restated as the opening numbered results of §3.4, p. 57, and appear in the chapter Summary, p. 72