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Introduction to Trigonometry
Introduction
Trigonometric Ratios
Practice: Ratios in a Right Triangle
Trigonometric Ratios of Some Specific Angles
Practice: Ratios of Special Angles
Trigonometric Identities
Practice: Trigonometric Identities
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Introduction to TrigonometryIntroduction

Reading time: ~15 min

Riya's class is on a school trip to the Qutub Minar in Delhi. Standing in the courtyard, their guide asks a question nobody expects: how tall is the Minar? Nobody may climb it, no measuring tape is long enough, and the guide wants the answer before lunch.

Riya has a simple angle-measuring tool, a clinometer. She sets it on the ground 42 m from the foot of the Minar — a distance she can pace out easily — and tilts its sighting tube upwards.

You hold the clinometer. Tilt the line of sight until it passes exactly through the top of the Minar. On the right, a small triangle with the same angle shows how far the line climbs for every 1 m it travels across.

Angle of the line of sight: ${a}°

At 60° the line of sight climbs about 1.73 m for every metre it travels across the ground. Over 42 m that adds up to 42 \times 1.73 \approx 72.7 m — and the Qutub Minar really is about 72.5 m tall. One angle and one easy distance gave a height nobody could reach. The number 1.73 belongs to the angle 60° alone; later in this chapter you will meet it again as tan 60°.

Right Triangles Hiding Everywhere

The Minar is not a one-off. In each of these three situations a right triangle hides in the scene, with one side we can measure and one side we cannot reach.

1. Students visiting the Qutub Minar want its height without measuring it directly (Fig. 8.1).

2. A girl sits on the balcony of her house on one bank of a river, looking down at a flower pot on the steps of a temple across the water. If she knows how high up she is sitting, can she work out how wide the river is (Fig. 8.2)?

3. A girl spots a hot air balloon at point A and runs indoors to tell her mother. By the time they both come out, it has drifted to point B. How high above the ground is B (Fig. 8.3)?

Tap each scene to find the right triangle hiding in it.

Every time, the triangle has a right angle at the ground, a side we can measure, and the side we want. Getting from the side we know to the side we want is the job of trigonometry, the mathematics of the .

Three Sides, One Measure

The word trigonometry is built from three Greek pieces: tri (three), gon (sides) and metron (measure). It is the study of how the of a triangle are related. Its earliest traces are found in Egypt and Babylon, where astronomers used it to estimate the distances of the stars and planets. Today engineers, surveyors, game designers and physicists use it every single day.

In this chapter you will define six trigonometric ratios of an acute angle in a right triangle, find their exact values for 0°, 30°, 45°, 60° and , and prove some trigonometric identities — equations in these ratios that hold for every angle. We stay with acute angles (plus 0° and 90°) throughout; the same ratios can be stretched to other angles in later classes.

Sina