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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 Ukrainian:

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:

, , , (earth surface), , .. (steel measuring tapes). (goniometrial instruments).

 

 

TEXT 2

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 pronuciation:

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 following 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 Ukrainian:

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) :

tgα=sinα/cosα

2) :

ctgα = cosα/sinα

TEXT 3

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 3115'10".

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. 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 Ukrainian:

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:

, 1/360 . 60 ; 60 .

 

 

TEXT 4





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