9. Some Applications of Trigonometry

4 topics54 min

Chapter 9 of NCERT Mathematics for Class 10: Some Applications of Trigonometry. Two sections — Setting up the sightline and Getting the height or the distance. Four videos, 54 min in all.

Worked answers to the 15 exercise questions in this chapter: Exercise 9.1

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Part 1

Setting up the sightline

13 minThe sightline, and the horizontal it gets measured againstAn angle needs two rays. The sightline is one. The second one is a choice, and taking it level THROUGH THE EYE is what puts a right angle into a field that had no triangle in it - and what leaves the answer short by exactly your own height.13 min · 12 parts13 minLooking up versus looking down: elevation and depressionElevation and depression are not two rules. They are ONE construction read twice, and the only thing that changes is which side of the level ray the target falls on. Which is why every depression problem is an elevation problem you have not relabelled yet.13 min · 12 parts

Part 2

Getting the height or the distance

14 minPicking the one ratio that links what you know to what you wantName the side you measured and the side you want, and the ratio is already chosen. Of the six names, two answer that question, two answer a different one, and two cannot be used at all - counted over 5184 trials, and it comes out two, two and two every time.14 min · 12 parts15 minTwo angles in one figure, and the pair of equations they hand youTwelve different towers all cast a shadow that reads sixty degrees. Exactly one of them also reads thirty from forty metres further back. That is what a second sighting is for: not a second measurement, a second equation.15 min · 13 parts

What you will be able to do

  • Identify, in a described or drawn situation, the observer's eye, the point of the object being aimed at, and the ray joining them
  • Classify a described sighting as an elevation or a depression by asking only whether the target sits above or below the eye
  • Letter a right triangle so that the known side, the wanted side and the known angle are all identifiable at a glance
  • Recognise from a problem statement that no length inside either triangle has been given, and that a second angle is therefore doing structural work

What this chapter assumes you already know

  • The six trigonometric ratios of an acute angle in a right triangle, and which side each pair names — carried straight in from Chapter 8, which this chapter opens by pointing back to
  • The exact values of the ratios at 30°, 45° and 60°
  • That a level line and a plumb line meet at a right angle
  • Reading a lettered figure: naming a segment by its two endpoints, and naming an angle by three letters with the vertex in the middle
  • Solving a one-unknown linear equation of the form (unknown)/(number) = value
  • Surds: leaving an answer as a multiple of √3 and, separately, turning it into a decimal with a stated approximation
  • Alternate angles at a transversal cutting a pair of parallel lines, from the earlier work on lines and angles
  • That opposite sides of a rectangle are equal, and that a rectangle's sides meet squarely

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.

  • "The angle is measured up from the ground."
  • "The angle sits at the object."
  • "The vertical side of the triangle is the height of the object."
  • "You can skip adding your height; it's rounding error."
  • "But Example 1 didn't add anything, so the rule is optional."
  • "The distance to the foot is the distance to the object."
  • "The drawing has to be to scale for the reasoning to hold."
  • "Depression is measured down from the vertical."