Trigonometric FunctionsIntroduction

You run the wheel. Drag Riya's seat round the rim. The purple spoke shows how far the wheel has turned, and the red line is her height above (or below) the axle. Every angle you visit also leaves a dot on the strip. Complete all three missions.

In Class 10 you would have used a right triangle: height = 10 × sin(angle turned). That works while the angle stays between 0° and 90°. But at 180° there is no triangle at all, and at 270° Riya is below the axle — a negative height that no triangle can produce. Yet her height is perfectly clear at every angle, and the dots on your strip line up into a smooth wave. This chapter builds the functions that describe that wave, for an angle of any size.

The word trigonometry comes from two Greek words, trigon and metron, and means "measuring the sides of a trianglesquarecircle". It began as a toolkit for geometricalgebraicarithmetic problems about triangles: ship captains used it to navigate, surveyors to map new land, engineers to build. Today it turns up far away from triangles — in seismology, in electric circuits, in the physics of atoms, in predicting ocean tides and in analysing musical notes.

In Class 10 you met the trigonometric ratios of acute angles as ratios of the sides of a right-angled triangle, proved some trigonometric identities, and used the ratios to find heights and distancesareasvolumes. Now we set the angle free from the triangle. It may be positive, negative or bigger than a full turn — exactly like the angles of Riya's wheel — and the ratios become functionsconstantsequations: one output for every input.

Born in India

Bhaskara I gave rules for the sine of angles larger than 90°. A sixteenth-century Malayalam work, the Yuktibhasa, contains a proof of the expansion of sin (A + B) — a formula you will prove yourself later in this chapter. Bhaskara II gave exact values for the sines of 18°, 36°, 54° and 72°. Much later, in 1813, the astronomer John Herschel suggested the symbols , for arc sin x and arc cos x.

Long before all of them, Thales (about 600 B.C.E.) is said to have found the height of a great pyramid in Egypt from its shadow. He compared the pyramid's height H and shadow S with the height h and shadow s of a staff, using

He is also said to have found the distance of a ship at sea from the sides of similarcongruentequal triangles. Height-and-distance problems solved with similar triangles appear in ancient Indian works too.

Over two and a half thousand years the questions grew from triangles to whole circles. In this chapter you will measure angles in a new unit, define six trigonometric functions for every angle, draw their graphs, and prove the formulas for sums and differences of angles.