PLANE TRIGONOMETRY.

ARTICLE VI

TRIGONOMETRICAL RATIOS OF THE SUM AND DIFFERENCE OF Two ANGLES.

In a previous article (see COMPETITOR for December and February, Nos. 5 and 7) we have shown how to find the trigonometrical ratios of certain angles, and also of half an angle, when those of the whole angle are known. We shall now show how the functions of an angle, which is the sum or difference of two or more angles, may be expressed in terms of the functions of these angles.

(a) For sin (A + B)

Let A and B, represented respectively by BAC, ABC, denote the angles whose trigonometrical ratios are known, and let the lengths of the perpendiculars from A and C on AB and BC, i.e., on c and a, produced if necessary, be p and P; and, to simplify the construction, let the angle (A+B) be less than a right angle.

because each side of this equation is an expression for double the area of the triangle ABC.

If B = A then sin (A + B) = sin 24 sin A cos A+ cos A sin A

(3) For cos (A + B)

Let CAB, CAD, denote the angles A and B respectively. From the point B draw BD, BE, perpendiculars to AM, AC, produced if necessary.

That is, aPp. AE-p. CE. (Each side of this equation is an expression for double the area of the triangle ABC.)

Cos (A+B) = cos 24 = cos A cos Asin A sin A

That is, cos 24 = cos A - sin' A. = 2 cos2 A-1=1-2 sin2 A.

Let the angle A be represented by BAD (see Fig 2), then the angle BAE represents (4- B). And sin (AB) = sin BAE= cos ABE, (because the angle E is a right angle).

But cos ABE = cos ABD. cos DBE-sin ABD. sin DBE.

Therefore sin (AB) = sin A cos B-cos A sin B

.. cos ABD = sin A

If B =

and sin ABD = cos A, the angle ADB being a right angle. (See Trig. Art. IV., COMPETITOR for November.)

A, then sin 0° 0. (We shall in a subsequent article refer to this expression.)

=

(8) For cos (4-B)

Same representation as in last paragraph. (See fig. 2.)

Cos (AB)=sin ABE = sin ABD, cos DBE + cos ABD sin DBE

We have thus established what are called the Four Fundamental Formula of Plane Trigonometry. With these the student should make himself thoroughly familiar; as much of what follows in the modern treatment of the science rests upon these formulæ. We shall repeat them here in a collected form.

Similarly from (7) and (8)

Cos (A+B) + cos (4B) = 2 cos A cos B
= 2 sin 4 sin B

And cos (4B) — cos (A + B) : Note: The smaller the angle the greater its cosine; hence we take cos (A to make the product plus.

This arrangement, which is very convenient in practice, may be expressed in words as follows:5. The sum of the sines of any two angles is equal to twice the sine of half their sum multiplied by the cosine of half their difference.

4

6. The difference of the sines of any two angles is equal to twice the cosine of half their sum multiplied by the sine of half their difference.

7. The sum of the cosines of any two angles is equal to twice the cosine of half their sum multiplied by the cosine of half their difference.

8. The difference of the cosines of any two angles is equal,in absolute magnitude,to twice the sine of half their sum multiplied by the sine of half their difference; the product being PLUS or MINUS, according as the cosine of the less angle is plus or minus.

Similarly the expressions in (1), (2), (3), and (4), so useful in changing the product of the sine and cosine of two angles, or the product of their cosines, or of their sines, into the sum or difference of the sines or cosines of these angles, may be stated in words as follows:

(1) The sine of the sum plus the sine of the difference of any two angles = twice the sine of the greater angles, multiplied by the cosine of the less.

(2) The sine of the sum minus the sine of the difference of any two angles of the greater angle multiplied by the sine of the less.

(3) The cosine of the sum plus the cosine of the difference of any two angles duct of their cosines.

(4) The cosine of the difference minus the cosine of the sum of any two angles duct of their sines.

twice the cosine

Functions of an angle which is the sum of three given angles A, B, C.

Sin (A+B+C) = sin {(A + B) + C}

sin (A+B) cos C + cos (A + B) sin C

(sin A cos B+cos A sin B) cos C + (cos A cos B-sin A sin B) sin C

twice the pro

twice the pro

Sin A cos B cos C+ sin B cos A cos C + sin C cos A cos B sin A sin B sin C

Suppose A, B, C, to be in descending order of magnitude, then we may state this in words as follows:

=

The sine of an angle which is the sum of three angles the sine of the greatest angle multiplied by the product of the cosines of the other two, plus the sine of the middle angle multiplied by the product of the cosines of the other two, plus the sine of the least angle multiplied by the product of cosines of the other two, minus the continued product of their sines.

Sin (a+B+2)

=

sin a cos B cos y + sin 3 cos a cos y + sin y cosa cos B Sin (++)

sin a sin ẞ sin 7.

= sin cos cos + sin cos cos + sin cos cos sin ◊ sin sin .

If CB= A; then sin (A+B+C) sin (A + A + A);

or sin 3A = sin A cos2 A + sin A cos2 A + sin A cos2 A - sin3 A.

= (cos A cos B

sin (A + B) sin C

sin A sin B) cos C- (sin A cos B + cos A sin B) sin C

=cos A cos B cos C-cos A sin B sin C-cos B sin A sin C-cos C sin A sin B.

With the same supposition as above, this may be stated in words as follows:The cosine of an angle which is the sum of three angles = the continued product of their cosines minus the cosine of the greatest angle multiplied by the product of the sines of the other two, minus the cosine of the middle angle multiplied by the product of the sines of the other two minus the cosine of the least angle multiplied by the product of the sines of the other two.

Cos (a + B + y) = cos a cos 8 cos y · cos a sin ẞ siny -
Cos (X + Y + Z) = cos X cos Y cos Z-cos X sin Y sin Z

cos B sin a sin y

cos Y sin X sin

Dividing the terms of this fraction separately by cos A cos B, we find

Wherefore the tangent of an angle which is the sum of two given angles tangents divided by the difference between unity and the product of their tangents.

=

the sum of their

that is, the tangent of twice an angle twice the tangent of the angle divided by the difference. between unity and the square of the tangent of the angle.

The tangent of an angle which is the difference of two given angles tangents divided by the sum of unity and the product of their tangents.

the difference of their