The circle (Fig. 143) is divided into four quadrants of 90 degrees. each, and by Definition 4 (page 85) each of these is a right angle. In trigonometry it is usual to consider the radius of a quadrant as unity, and as a line identical with the horizontal arm of the quadrant moves in an upward direction towards the vertical arm a c, Fig. 144, so the angle formed by this line produces certain functions which, for simplicity, are considered in the terms of the angle so formed, usually called the angle A. Thus Fig. 145 shows the angle a equal to 30 deg.; Fig. 146, the angle a equal to 45 deg. ;

Fig. 147, the angle A equal to 60 deg.; and so a diagram may be constructed to represent an angle which is any fractional part of 90 deg.

It may be well here to introduce and explain the trigonometrical canon or diagram (Fig. 148), which shows the different trigonometrical functions in terms of the angle a to the radius 1.

COMPLEMENT

=

Now here, for simple illustration, I have taken the angle A as 45 deg.

The trigonometrical functions of the angle A are as follows: The SINE, CO-SINE, TANGENT, CO-TANGENT, SECANT, and Co-SECANT, with the VERSINE and CO-VERSINE, but the two latter do not enter largely into the consideration of the solution of triangles.

Now Fig. 149, illustrating the functions of an angle of 30 deg., shows by the strong lines certain positive functions of that angle, such as the sine, secant, and tangent; whilst the extended dotted lines, and dotted lines, show the complementary functions of the same angle, as the co-sine, co-secant, and co-tangent.

RATIOS OR FUNCTIONS.

*

91

Here I should explain that the complement of an angle is equal to its difference from 90 deg., so that 60 deg. is the complement of 30 deg.

The supplement of an angle is equal to its difference from 180 deg., so that the supplement of 30 deg. is 150 deg.

By referring to Figs. 149 and 150 it will be seen that in the former case the sine, secant, and tangent are much less than the co sine, co-secant, and co-tangent (which are shown by dotted lines) by reason of the angle being small; whilst in Fig. 150 it will be seen that the sine, secant, and tangent are greater than are the co-sine, co-secant, and co-tangent; and going back to Fig. 148, we have the sine equal to the co-sine, the tangent equal to the co-tangent, and the secant equal to the co-secant.

Trigonometrical Ratios or Functions.-1. Sine.-The sine of an arc is a perpendicular let fall from the extremity of one radius to the other, as E F (Figs. 148, 149, and 150).

2. Tangent. The tangent is a perpendicular line drawn from the extremity of the radius to meet the other produced, as B D.

3. Secant.-The secant is that radius which forms 'the angle produced until it meets the tangent, as a D.

4. Cosine.-The cosine is a line drawn parallel to that part of the radius between its centre and the foot of the sine.

5. Cotangent.-The cotangent is a horizontal line, commencing at the termination of the quadrant, and terminating on the

The difference between an acute angle and a right angle is called its complement (i.e. the angle lacking to complete or fill up the right angle).

radius ▲ E produced, in D (Fig. 148), D' (Fig. 149), and D (Fig. 150).

6. Cosecant. The cosecant is one of the radii produced until it intersects the cotangent in D (Fig. 148), and D' (Fig. 149 and 150).

7. Versed Sine.-The versed sine is that portion of one of the radii between the foot of the sine and the arc on E B.

8. Coversed Sine.-The coversed sine is that portion of the perpendicular between the cosine and the quadrant, as G c.

9. Chord.-The chord of an arc is a line joining the extremities of the arc.

I should like here to explain what may appear to be an anomaly, viz. why the lines GE (COS A), C D' (cot A), and A D' (cosec A) (Fig. 149), should be the complementary to the functions of the angle a. But I hope the following will elucidate the matter. We have found (page 91) that the complement of an angle is the angle lacking to complete or fill up the right angle; and by reference to Fig. 149 it will be seen that the line G E bears the same relation to the angle E A C as E F does to the angle A or E A B, consequently & E must be the sine of the angle E A C. Thus what is the sine of an angle (less than 90 deg.) is the cosine of the remaining angle or complement, and vice versâ. The line c D' bears the same relation to the angle E A C as D B bears to the angle E A B, therefore what is the cotangent of the angle E AB is the tangent of the angle E A C; and the same equally applies to the secant and cosecant.

These trigonometrical functions are abbreviated as follows:

= The sine of the angle a.

:

Sin A

Cos A

Relation of Hypotenuse to the other Sides of Right-angled Triangle. Perhaps it may be better to refer to the 47th proposition of Euclid, which states the theorem: "In any right-angled triangle, the square which is described on the side subtending the right angle is equal to the sum of the squares described on the sides which contain the right angle " (Fig. 151).

By this proposition the sum of the squares on the sides A and B is equal to that on the side c; in other words, taking another form of a right-angled triangle, as Fig. 152

Now in the preceding descriptions of the various trigonometrical functions, I have shown that they all have reference to the angle A of the triangle в A c, a portion of the first quadrant (see Fig. 153),

which is placed in the centre of the circle called the circle of reference.

B

93

We will now consider the functions of the angle A (в A c) in terms of the sides of the triangle A C B. We have seen (Figs. 149, 150) that the functions are the ratios borne by certain lines to the radius; and as a ratio or proportion may always be expressed in the form of a fraction, the functions may be obtained by dividing these lines by the radius. Now, so long as the angles of a triangle remain unchanged, the ratios of the sides of that triangle remain unchanged; hence, comparing Fig. 154 with Fig. 149 or Fig. 150, we are able to express the functions of the angle a in terms of the sides A B, B C, C A.

BCA B COS B; A C A B sine B; A B = B C sec B.

B = complement of a = 90

A+B+C = 180°.

A.

I may explain, by reference to Fig. 148, that the tangent, cotangent, secant, and cosecant appear therein much longer than the lines E F, A F, and E A, which correspond with the lines B C, A c, and A в in Figs. 153 and 154; and my reason for referring to it is to show that, as these lines are simply ratios to the radius, so what in Fig. 148 is the tangent of a, viz. is exactly the same

BD

A B

B C ratio as A C

in Figs. 158 and 154, or as follows: