There will be two solutions from the two values of +ß given in (2).

C

Find from the tables the value of cos ß. Next find from the tables the magnitude of the angle a whose sine: cos ẞ, and we get

sin(+6)=sin α,

=

α

:: &+ß = nñ +(-1)"α . . (Art. 38)

.. β

where n is zero or any positive or negative integer.

In order that the solution may be possible, it is necessary

NOTE. This example might have been solved by squaring both sides of the equation; but in solving trigonometric equations, it is important, if possible, to avoid squaring both sides of the equation.

(6) is the complete solution of the given equation (3), while (5) is the solution of both cos 0 = k sin 0, and also of cos ek sin 0. Therefore by squaring both members of an equation we obtain solutions which do not belong to the given equation.

* The minus sign is written thus to denote that it belongs to the natural number and does not affect the logarithm. Sometimes the letter n is written instead of the minus sign, to denote the same thing.

0

= 17.5($< 180°).

π

3. Solve 2 sin @ +2 cos 0=√2. Ans. −7+n+(−1) "T

π

π

3

4

4

which determines x+a, and therefore x.

If we introduce an auxiliary angle, the calculation of equation (3) is facilitated.

Thus, let m=tano; then we have by [(14) of Art. 61]

gives the logarithmic solution.

The logarithmic solution of the equation

sin (α- x)= m sinx

is found in the same manner to be

sin (106°+x)=-1.263 sin x (x < 180°).

Example. Solve tan (23° 16' + x)= .296 tan x.

log tan & = log m = log(.296) = 1.47129.

.. φ = 16° 29'.3.

$ — 45° — — 28° 30'.7; log cot ( — 45°) = 10.26502 –

Example.-Solve tan (65° + x) tan x = 1.5196 (« < 180°).

for m and x, the other four quantities, 0, 4, a, b, being

known.

Expanding (1) and (2) by (Art. 44), we get

m sin cos x + m cos 0 sin x = a

m sin cos x + m cos o sin x = b

(3)

(4)

Multiplying (3) by sino and (4) by sin 0, and subtracting the latter from the former, we have

m sin x (sin cos 0 — cos & sin 0) = a sin & − b sin 0.

To find the value of m cos x, multiply (3) and (4) by cos and cos 0, respectively, and subtract the former from the latter.

Thus

Having obtained the values of m sina and m cos x from (5) and (6), m and x can be calculated by Art. 83.