Lesson 1 of 13 · 8 min

Angles and degree measure

NCERT §3.2.1

Meera boards the giant wheel and her cabin starts level with the hub, on the east side. A minute later the wheel has carried her to the very top. How do we say exactly how far the wheel has turned, and what if the operator runs it backwards?

The story this chapter follows: The giant wheel at the town fair

Meera and her friends ride the giant wheel at the town fair. Picture it with a hub 9 m above the ground, a radius of 7 m, and one slow anticlockwise revolution every 4 minutes. The numbers are chosen for easy arithmetic. Put the hub at the origin with the ray pointing east as the starting position, and the wheel supplies every angle, arc and trigonometric value in this chapter.

The lesson in notes

In short

An angle measures how far a ray has turned about its starting point. The ray's starting position is the initial side, its final position is the terminal side, and the fixed point it turns about is the vertex.

Direction carries a sign: a turn in the anticlockwise sense gives a positive angle, and a turn in the clockwise sense gives a negative angle.

One natural unit is the revolution, a full turn that brings the ray back to its initial side. It suits large, repeated turns, such as a wheel spinning a number of revolutions each second.

One degree, 1°, is the angle of a turn through 1/360 of a revolution, so a full turn is 360°, a half turn 180° and a quarter turn 90°.

Degrees are split sexagesimally: 1° = 60′ (minutes) and 1′ = 60″ (seconds), so 1° = 3600″. For example, 40° 15′ = 40.25°.

Angles are not limited to 0° to 360°. A turn of 420° is one full revolution followed by 60° more, and −30° is a 30° turn clockwise; 420° and 60° end on the same terminal side.