Trigonometry and its application

Trigonometry is a branch of mathematics that deals with angles and sides of triangles and their relations to one another. Otherwise we can say that trigonometry is the set of methods and procedures required to solve problems concerning trian­gles when angles of the triangles are involved. It is especially useful in numerous calculations -connected with accurate machine work. It is also very useful to the surveyor, draughts­man and, in fact, is used in all sorts of engineering work. The carpenter with his steel square makes use of trigonomet­ric relations to find the length of one side of hip roof.

Many problems about alternating current can be solved by using trigonometry. In meteorology, the height of a balloon above the surface of the earth is determined by the use of trigonometry.

Trigonometry has applications in surveying, navigation, construction work and many branches of science. It is particularly essential for most branches of mathematics and physics.

EXERCISES

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

calculation, relation, ratio, angle, triangle, root, roof, balloon, use, cube, deal, meeting.

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

to be useful to, in engineering work, otherwise, branch of mathematics, branch of physics, to deal with angles, to deal with; triangles, to deal with relations.

III. Answer the following questions:

1. What does trigonometry deal with? 2. In what work is trigonometry especially useful? 3. For what purpose does the carpenter make use of trigonometric relations? 4. What is determined in meteorology by the use of trigonometry?

IV. Translate into Russian:

A branch of mathematics dealing with the relationship between the sides and angles of triangles is called trigonom­etry. It is defined as the branch of mathematics using the fact that numerous problems may be solved by the calculation of unknown parts (sides and angles). The solution of such problems is greatly assisted by the use of the trigonometrical ratios or functions.

V. Translate into English:

С помощью тригонометрии решаются многие измери­тельные задачи на местности, как например, вычисление расстояний между различными пунктами земной поверх­ности (earth surface), вычисление высоты данного предмета, составление карт и т.п. Измерение небольших расстояний производится при помощи стальных измерительных лент (steel measuring tapes). Измерение углов производится при помощи угломерных инструментов (goniometrial instru­ments).

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TRIGONOMETRIC FUNCTIONS

Trigonometry is based on certain "functions" of angles. A function is quantity that depends on another quantity for its value. Any quantity that depends upon an angle for its value is the function of that angle. If a right triangle is con­structed, having a certain angle at one corner, there will be certain definite relations between the sides of this triangle.

These ratios are six in number and are called the trigono­metric functions.

In any right triangle, we call the two lines that form the right angle the sides, while the line opposite the right angle is called the hypotenuse.

In speaking of the angle PON in the triangle PON (Fig. 30) the side NP is called the opposite side, while the side ON is called the adjacent side. The ratio of the opposite side to the adjacent side is called the tangent of the angle. The tan­gent is abbreviated tan. Hence, tan <NOP=NP/ON. The ratio of the adjacent side to the opposite side is called the cotangent (abbreviated cot). Cot <NOP=ON/NP, the word cotangent is an abbreviation or shortening of the word: "complementary tangent" or "tangent of the complementary angle". The tan­gent of any angle is the cotangent of its complement, and the cotangent of any angle is the tangent of its comple­ment.

In any right triangle, the sine of either acute angle is the ratio of the side opposite the angle to the hypot­enuse. In writing "the sine of the an­gle PON" in an equation or formula, it would be abbreviated sin PON. Hence, sin <PON=PN/OP.

In any right triangle, the cosine of either acute angle is the ratio of the adjacent side to the hypotenuse. In writing "the co- 'sine of the angle PON" in an equation or formula, it would be abbreviated cos PON. Hence, cos <PON=ON/OP.

The secant of the angle is the reciprocal of the cosine; that is, it is reverse ratio of the cosine. In a right triangle, the secant of an angle is the ratio of the hypotenuse to the side adjacent to the angle. The secant is abbreviated sec.

The cosecant is the reciprocal of the sine; that is the re­verse ratio, being the ratio of the hypotenuse to the side opposite the angle. The abbreviation for the cosecant is csc.

EXERCISES

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

right, height, identify, satisfy, sine, cosine, combine, secant, cosecant, tangent, cotangent.

II. Add suffixes to the words given below and translate them into Russian:

-wise: clock, counterclock, other;

-tion: relate, opposite, abbreviate, definite;

-ing: write, call, construct, shorten.

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

to be based on, to depend on, for its value, is constructed, five in number, hypotenuse, opposite side, adjacent side.

IV. Answer the wollowing questions:

1. What is a trigonometric function? 2. What lines in any right triangle are called sides? 3. What is called the tangent of the angle? 4. What is called the cotangent? 5. What is the sine of an angle? 6. What is the cosine of an angle? 7. What is the secant of an angle? 8. What is the cosecant of an angle?

V. Translate into Russian:

We find that the values of the right triangle ratios depend only on angles and that to each different angle there corre­sponds a different set of these values. Any number or quantity which is related in this way to another number or quantity is called a function of that second number or quantity. There­fore, the right triangle ratios are functions of the acute an­gles of the triangle.

VI. Translate into English:

1)Тангенсом угла а называется отношение синуса угла а к косинусу этого угла:

tgα=sinα/cosα

2)Котангенсом угла а называется отношение косинуса угла а к синусу этого угла:

ctgα = cosα/sinα

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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 thecircle. 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.

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