SECTION 2.

TRIGONOMETRY.

DEFINITIONS.

1. Geometry shows how to construct a triangle and how to find the unknown parts by actual measurement. Trigonometry shows how to compute the unknown parts from the numerical values of the known parts.

In any triangle there are six parts: three sides and three angles. These six parts are so related that any three, provided one of them is a side, determine the shape and size of the triangle. 2. Trigonometry, then, is that

branch of mathematics which treats of the numerical computation of angles and triangles.

CE G M

Fig. 1.

3. In deducing rules or formulas to be used in computations, we use what are called functions of the angles. This word function is used in mathematics to denote a quantity which depends on some other quantity for its value. Thus the area of a circle depends upon the radius of a circle for its value; hence the area of a circle is a function of the radius.

4. Let MAN be any acute angle. From B, D and F, any points in A N, draw lines perpendicular to A M, forming the right triangles BCA, DEA and F G A. These right triangles are similar, since they are mutually equiangular. The homologous sides are then proportional. Therefore:

These ratios, which depend upon the magnitude of angle A

for their values, are functions of A.

and they are defined and named as follows:

5

There are six such ratios

Fig. 2.

Example.

6 Fig. 3.

Take the right triangle A C B, Fig. 2, with the

sides 3, 4, 5. Then from our definitions:

5.

When in Art. 13 to Art. 17 we take up the solution of right triangles, we shall use a triangle lettered as in Fig. 3. A and B are the acute angles, C the right angle, and the opposite sides are represented by the corresponding small letters. That is, the side opposite angle A is a, that opposite B is b, etc.

The six ratios are called the Trigonometric Functions of angle A.

6. The following definitions must be committed to memory: The sine of an angle is the ratio of the opposite side to the hypothenuse. The cosine of an angle is the ratio of the adjacent side to the hypothenuse. The tangent of an angle is the ratio of the opposite side to the adjacent side.

The cotangent of an angle is the ratio of the adjacent side to the opposite side.

The secant of an angle is the ratio of the hypothenuse to the adjacent side.

The cosecant of an angle is the ratio of the hypothenuse to the opposite side.

7. If we apply these definitions to angle B we have the following:

By inspection of the functions of A and B (which are complementary angles) we see that:

EXAMPLES FOR PRACTICE.

Calculate the value of the six functions of angle A in the right triangle AC B, whose sides a, b, c are respectively:

Given one function of an angle to find the other functions. Suppose we have given sin A

Find the values of the five remaining functions, when

Measurement of Angles. As we learned in "Mensuration," angles are usually expressed in degrees, minutes and seconds. A degree is a ninetieth part of a right angle; a minute is one-sixtieth of a degree, and a second is one-sixtieth of a minute. The symbols used to denote degrees, minutes and seconds are ". Thus forty-six degrees fifteen minutes and six seconds is written 46° 15' 6". In adding two angles, seconds are added to seconds,

minutes to minutes, and degrees to degrees; also, in subtracting two angles, seconds are subtracted from seconds, minutes from minutes, and degrees from degrees.

Example 1.- Add 67° 45' 16" and 32° 28' 52".

Solution:

67° 45′ 16′′

32° 28' 52"

99° 73' 68′′

But 68" 1' 8" and 73' — 1° 13'.

Therefore 99° 73' 68" 100° 14′ 8′′ Ans.

Example 2.- Subtract 35° 28' 14" from 119° 14′ 25′′. Solution:

119° 14' 25"

1

35° 28' 14"

Since 28' cannot be taken from 14' we borrow 1° (= 60') from 119° and write the angles

Example 3.— In a right triangle one acute angle is 16° 25′ 13′′, find the other acute angle.

We know that the sum of the acute angles of a right triangle is equal to one right angle. We write 90° as 89° 59′ 60", then subtracting the given angle,

89° 59′ 60"

16° 25′ 13′′

73° 34′ 47′′ Ans.

EXAMPLES FOR PRACTICE.

Ans. 180° 12′ 20′′.

Ans. 96° 11′ 49′′.

Ans. 39° 58' 53".

1. Add 176° 59′ 24′′ and 3° 12′ 56′′. 2. Add 56° 28′ 32′′ and 39° 43′ 17". 3. From 84° 19' 27" take 44° 20' 34". The Functions of 45°. In order to of 45° take a right isosceles triangle A CB about the right angle being equal to unity. Then

10.

find the functions (Fig. 5), each side