Chapter 9: Some Applications of Trigonometry

The previous chapter built the tools , sin, cos, tan, identities. This chapter uses them. The classic application is heights and distances: from a known distance and a measured angle, compute the height of a building, the width of a river, or the position of an aircraft.

For board exams, this chapter is famously formulaic , once you sketch a clean diagram, the trig is straightforward. Expect $3$- to $5$-mark word problems on towers, ladders, kites, and observers with two angles of elevation or depression. Two-angle problems (with two unknowns) often need two equations and a substitution.

In the real world, trigonometry's applications are enormous: surveying (which built the Indian railway system), astronomy (measuring distances to stars by parallax), GPS (your phone's location), and architecture (load-bearing analysis). All grow out of the right-triangle reasoning here.

What's inside

• Angles of elevation and depression , definitions and conventions.
• Single-angle problems , one observer, one object.
• Two-angle / two-object problems , multiple observers or one observer with two angles.
• Indirect-measurement set-ups , kites, boats, towers, slopes.
• Common pitfalls and clean-diagram habits.

Key results / Formula card

Always:

• Draw a clean diagram before any algebra.
• Mark the observer's eye, the object's tip, the horizontal line, and the line of sight.
• The angle of elevation is measured upward from the horizontal; the angle of depression is measured downward.
• For a right triangle with horizontal $b$, vertical $h$, and angle of elevation $\theta$: $\tan\theta = \frac{h}{b}, \quad \sin\theta = \frac{h}{\text{line-of-sight}}, \quad \cos\theta = \frac{b}{\text{line-of-sight}}.$

Sometimes you'll need the standard angles' values from chapter 8 , keep that table handy.