SM2.4 Heights and distances
You can turn a described scene into a right-triangle diagram with angles of elevation or depression, and solve for the height or distance asked.
Topics shown in module order.
This chapter is trigonometry pointed at the world: a tower seen from the ground, a bird seen from a balcony, a boat seen from a cliff. Every question is the same three moves. Draw the scene as one or two right triangles, mark the angle of elevation or depression where the observer stands, and let the ratios from the last chapter do the arithmetic. The drawing is the answer; the calculation only finishes it.
Work through these
For several described scenes, draw the diagram only: triangles, known sides, the angle, and what is asked
Nearly every lost mark in this chapter is a wrong diagram, not a wrong ratio. Practising the drawing as its own step, with no solving allowed, trains the half of the skill the textbook exercises rush past.
Solve single-triangle problems, choosing the one ratio that connects what you know to what is asked
With one side known and one wanted, exactly one of tangent, sine or cosine does the job in a single step. Choosing it deliberately, rather than writing all three and hoping, is what speed in this chapter is made of.
Solve two-triangle problems, where the observer or the object gives you a pair of angles
Two angles from one baseline, or one object seen from two places, produce two equations that meet in the middle. These are the examination's favourite form of the chapter, and they need the equations written before any numbers move.
Check one answer for plausibility against the scene: a flagpole should not come out taller than a hill
The arithmetic here produces confident nonsense when the diagram was wrong, and only the scene itself can tell you. A ten-second plausibility check is the habit that separates practised students from lucky ones.
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