3. Trigonometric Functions

11 topics3 h

Chapter 3 of NCERT Mathematics for Class 11: Trigonometric Functions. Three sections — Measuring an angle, From ratios in a triangle to functions on the line and Identities for angles built out of other angles. 11 videos, 3 h in all.

Worked answers to the 52 exercise questions in this chapter: Exercise 3.1 · Exercise 3.2 · Exercise 3.3 · Miscellaneous Exercise

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Part 1

Measuring an angle

12 minAn angle as an amount of turning, and what its sign recordsA corner cannot open past one full turn, or by less than nothing. Redefine an angle as a record of turning, and 420° stops being nonsense — one revolution, and more.12 min · 12 parts12 minWhy arc divided by radius is the measurement the mathematics prefersThree hundred and sixty is a number somebody chose; no circle demands it. Turn until the swept arc equals the radius, and the circle supplies its own unit instead.12 min · 13 parts17 minThe exchange rate between the two units, and the arc-length rule it buysA half turn measures a hundred and eighty in one unit and pi in the other — one equation, divided whichever way a question needs. What it buys works only in radians.17 min · 14 parts

Part 2

From ratios in a triangle to functions on the line

16 minCoordinates on the unit circle extend the ratios to every real numberA right triangle cannot ask for the sine of a hundred and twenty degrees — every angle in it besides the right angle is under ninety. A circle, with the angle at its centre, can.16 min · 14 parts14 minWhich of the six stays positive where, and what happens at the quarter turnsA table of twenty-four pluses and minuses looks like twenty-four things to learn. It is two: the signs of one point's own coordinates, and nothing more than that.14 min · 13 parts13 minWhere each function is defined, what values it reaches, and how it repeatsSine and cosine repeat every full turn; tangent and cotangent already repeat after half of one. Both facts were decided the moment the definition went on the circle.13 min · 14 parts

Part 3

Identities for angles built out of other angles

15 minOne distance calculation on the unit circle yields the cosine of a differenceGuess that a sum's cosine is the sum of the cosines, and two quarter turns kill it: nothing plus nothing against minus one. It survives by luck at eighteen of 576 pairs tried.15 min · 14 parts15 minShifts by a quarter, a half and a whole turn all fall out of the same two resultsA quarter turn added to an angle swaps its sine and cosine; a half turn never does. Both are one expansion — and which coordinate happens to vanish at the turn chosen.15 min · 14 parts13 minThe tangent and cotangent versions, and the angles they refuse to coverFour identities arrive with small print naming angles they refuse to cover. That print is not a warning — it is the division inside the proof, left exactly where it happened.13 min · 13 parts16 minSetting the second angle equal to the first gives the double and triple angle rulesDoubling an angle does not double its cosine. At thirty degrees the cosine is root three over two; twice that exceeds one, which no circle coordinate ever reaches.16 min · 15 parts16 minTrading a sum of two ratios for a product, and back againA sum of two sines cannot be cancelled or factored — nothing sits outside a bracket. Turn it into a product instead, and a common factor is suddenly there to use.16 min · 14 parts

What you will be able to do

  • State what a rotation-based angle records that a corner-angle cannot
  • State the definition of one radian in terms of an arc on a unit circle
  • Derive both conversion multipliers from the single half-turn equation
  • Explain why the ratio definition cannot be applied beyond an acute angle
  • Derive the sign of any of the six functions in any quadrant from the coordinates alone
  • State the domain of each of the six functions and justify it from the definition
  • State what §3.4 is allowed to assume before result 3 is proved
  • Derive result 5 by substituting a quarter turn into result 4
  • Derive result 10 as a quotient of results 7 and 3, and name the divisor used
  • Derive all four faces of the doubled cosine and say which identity converts each into the next
  • Produce the four raw equations by adding and subtracting the expansions in pairs

What this chapter assumes you already know

  • Rays, endpoints, and the ordinary corner-angle of Classes VI–X
  • The trigonometric ratios of an acute angle as side quotients in a right triangle, which §3.1 explicitly takes as the starting point
  • Reading a fraction of a whole, since a degree is defined as a fraction of a complete turn
  • Place value and the idea of a base other than ten, for the sixtieths that make minutes and seconds
  • How to get a circle's way round from its radius, the constant being 2π
  • That π is a real number a little larger than 3, and irrational
  • Proportion: what it means for two quantities to be in a fixed ratio
  • The real number line, including negative numbers and irrationals

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.

  • "An angle cannot be more than 360°."
  • "A negative angle is somehow less than nothing."
  • "The picture tells you the angle."
  • "40° 20′ means 40.2°."
  • "Revolutions are a childish unit."
  • "Trigonometry is about triangles."
  • "A radian is just 57.2957…°, so it is as arbitrary as a degree."
  • "The definition only works on the unit circle."