8. Introduction to Trigonometry

8 topics2 h

Chapter 8 of NCERT Mathematics for Class 10: Introduction to Trigonometry. Three sections — Six ratios from one acute angle, The angles you are expected to know and Identities. Eight videos, 2 h in all.

Worked answers to the 19 exercise questions in this chapter: Exercise 8.1 · Exercise 8.2 · Exercise 8.3

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Part 1

Six ratios from one acute angle

13 minWhich side is opposite and which is adjacent depends on the angle you pickA right triangle has three sides and only one of them - the hypotenuse - has a name of its own. Which leg is 'opposite' and which is 'adjacent' is decided by the angle you choose to stand at, not by the drawing.13 min · 13 parts14 minDefining sine, cosine and tangent, then their three reciprocalsSix ratios get names in this topic, and only three of them are new. The other three are the first three turned upside down - and the prefix 'co-' does NOT tell you which goes with which.14 min · 14 parts15 minWhy enlarging the triangle leaves every ratio unchangedWriting 'sin A' claims the number depends on the angle alone - but you needed a triangle to work it out, and the angle did not pick one. Similar triangles are what close that hole, and the whole proof is one cancellation.15 min · 14 parts13 minGiven one ratio, reconstructing the other fiveTell me one of the six ratios and I will tell you the other five. It sounds like too little to go on - and the letter you write down to make it work is one you will never solve for, because it was only ever there to cancel.13 min · 13 parts

Part 2

The angles you are expected to know

14 minSqueezing 30°, 45° and 60° out of two special trianglesThe sine of thirty degrees is exactly one half. The sine of thirty one degrees is a decimal that never stops. Nothing about the numbers is doing that - what thirty has is a triangle you can build, and there are only two such triangles in the whole topic.14 min · 13 parts14 minThe extreme cases at 0° and 90°, and the ratios that stop being definedThere is no right triangle with an angle of zero in it, and none with a second right angle. So the two columns at the ends of the table were never measured off anything - they were chosen. This is why the choice is forced, and why four cells have no value at all.14 min · 13 parts

Part 3

Identities

13 minDividing Pythagoras through by the hypotenuse squaredThe first identity of trigonometry is not a new fact and it is not something to memorise. It is Pythagoras, with every term divided by the square of the longest side. The lengths cancel, the triangle disappears, and what is left is a statement about the angle alone.13 min · 12 parts15 minTwo more identities from the same equation, and the angles they hold forThere are not three identities to memorise. There is one equation and three sides you could divide it by. And the angle each identity is NOT claimed at turns out to be exactly the angle where the side it divided by has collapsed - so the range is not fine print, it is the receipt for the division.15 min · 13 parts

What you will be able to do

  • Identify the hypotenuse of a right triangle from the position of the right angle alone, without measuring
  • Write the definitions of sine, cosine and tangent for a named acute angle of a right triangle in terms of opposite, adjacent and hypotenuse
  • Explain why "the sine of A" needs justification before it can be used as notation
  • Read a given ratio as a statement about the proportion of two named sides
  • Explain why exact values are available for some angles and not for others
  • Explain why 0° and 90° cannot appear as the working angle in an actual right triangle
  • Distinguish an identity from an equation that holds only for particular values
  • Derive the identity relating the tangent and the secant by dividing Pythagoras by the square of the adjacent leg

What this chapter assumes you already know

  • What a right triangle is, and that its two non-right angles are both acute
  • That the hypotenuse is the side facing the right angle, and is the longest side
  • Reading a labelled figure: vertex letters, the small square marking a right angle, and naming a side by its two endpoints
  • Ratio of two lengths, and that a ratio of two lengths carries no unit
  • The angle sum of a triangle, enough to see that the two acute angles of a right triangle add to 90°
  • Writing a ratio of two lengths as a fraction, and reducing it
  • What a reciprocal is, and that a reciprocal is undefined when the original is zero
  • Dividing one fraction by another, and cancelling a common factor top and bottom

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.

  • "BC is the opposite side, full stop."
  • "Adjacent just means touching the angle."
  • "The hypotenuse also changes when you switch angles."
  • "Opposite means the longest side."
  • "You can call the sides opposite and adjacent as soon as you see a right triangle."
  • "These pictures are about towers and rivers."
  • "cosec is short for cosine, so it pairs with secant."
  • "There are six independent things to learn."