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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