Measurement of angles

Degrees. There are common system for the measurement of angles. In one the degree is the unit of measurement.

The angle of one degree is the angle which requires 1/360 of the rotation needed to obtain one complete revolution.

Thus a complete revolution is divided into 360 equal parts called degrees. Each degree is divided into 60 equal parts called minutes, and each minute — into sixty equal parts called seconds. The symbols º ,', " are used to denote degrees,

minutes and seconds respectively. Thus angle of 31 degrees, 15 minutes and 10 seconds may be written 31°15'10".

Radians. In the second system used for measurement of angles, the radian is the unit of measure.

A radian is the measure of an angle which, placed with its vertex at the centre of any circle, subtends on the circum­ference an arc equal in length to the radius of the circle. Thus if we take a circle with centre at 0 and radius r and from a point A on the circum­ference measure an arc AB of length r, the angle AOB is by defi­nition an angle of 1 radian (Fig. 31). We may say that the length of an arc of a circle is equal to the radius of the circle multiplied by the meas­ure in radians of the angle subtend­ed by the arc at the centre of the circle. To convert degrees to radi­ans, divide the number of degrees by 180/π, or multiply by π/180 to convert radians to degrees, multiply the number of radians by 180/π.

Exercises

I. Read the following words paying attention to the pronun­ciation:

degree, meet, complete, coefficient, coincide, outside, arc, part, branch.

II. Make up sentences of your own using the words and expres­sions given below:

in one degree, to use for, the unit of measurement, con­vert, to express an angle, in the system, equal in length to, subtended by.

III. Answer the following questions:

1. What units of measurement of angles do you know? 2.What is called a degree? 3. Into how many parts is the degree divided? 4. How do we measure angles by using the radian? 5. How can degree be converted into radians?

IV. Translate into Russian:

Circumference of any circle is divided into 360 equal parts and lines are drawn from the centre of the circle through each point of division; the angle between any two successive ones of these radial lines is one degree. For measuring very small angles, the degree is divided into 60 equal parts each of which is called one second of angle. There are thus 21,600 minutes in a circle, 3,600 seconds in one degree, and 1,296,000 seconds in a circle.

V. Translate into English:

На практике углы часто измеряют в градусах, прини­мая за единицу измерения 1/360 часть полного оборота. Для измерений большей точности, градус делится на 60 равных частей — минуты; минута делится на 60 равных частей — секунды.

TEXT

Functions of complementary angles

In trigonometry it is occasionally convenient to speak of the cosine, the cotangent and the cosecant as the complemen­tary functions or co-functions of the sine, the tangent and the secant respectively. Conversely, the sine, the tangent and the secant are called the co-functions of the cosine, the co­tangent and the cosecant. Recalling that two angles are said to be complementary¹ if their sum is 90°, we shall prove the theorem that a trigonometric function of an angle is equal in value to the co-function of its complementary angle.

If two positive acute angles are known to have² a trigo­nometric function of one angle equal to the trigonometric co-function of the other, then the angles are complementary.

Notes:

¹ angles are said to be complementary — зд. углы счи­таются дополнительными

² if two positive acute angles are known to have — если известно, что у двух положительных острых углов