CHAPTER V.

SOLUTION OF TRIGONOMETRIC EQUATIONS.

82. A Trigonometric Equation is an equation in which the unknown quantities involve trigonometric functions.

The solution of a trigonometric equation is the process of finding the values of the unknown quantity which satisfy the equation. As in Algebra, we may have two or more simultaneous equations, the number of angles involved being equal to the number of equations.

EXAMPLES.

=

1 1. Solve sin e

2 This is a trigonometric equation. To solve it we must 1

1 find some angle whose sine is We know that sin 30°

2 Therefore, if 30° be put for 0, the equation is satisfied.

.:0= 30° is a solution of the equation.
.: 0=nn +(-1)".

(Art. 38) 6

5 2. Solve cos 0 + sec 0

2 The usual method of solution is to express all the functions in terms of one of them.

1 Thus, we put - for sec 0, and get

=

cos

This is an equation in which 0, and therefore cos 6, is unknown. We proceed to solve the equation algebraically just as we should if x occupied the place of cos e, thus:

The value 2 is inadmissible, for there is no angle whose cosine is numerically greater than 1 (Art. 21).

1 .. cos 0

2

=

Therefore one value of A which satisfies the equation is 60°.

3. Solve cosec 0 - cot20+1= 0. We have cosec 0 - (cosec 6 -1)+1=0., (Art. 23)

cosec- cosec 0 = 2.

Therefore 30° is one value of 0 which satisfies the equation.

Find a value of 0 which will satisfy the following equations : 4. cos 0 = cos 2 A.

Ans. m. 5. 2 cos 0 = sec 0.

= a

83. Solve the equations
m sin $

(1) m cos o=b .

(2) where a and b are given, and the values of m and $ are required.

Dividing (1) by (2), we get

a

tand

m =

sin •

b which gives two values of ¢, differing by 180°, and therefore two values of m also from either of the equations

b

COS • The two values of m will be equal numerically with opposite signs.

In practice, m is almost always positive by the conditions of the problem. Accordingly, sin • has the sign of a, and cos & the sign of b, and hence $ must be taken in the quailrant denoted by these signs. These cases may be considereil as follows:

(1) Sin $ and cos o both positive. This requires that the angle • be taken in the first quadrant, because sin $ and cos are both positive in no other quadrant.

=

(2) Sin positive and cos o negative. This requires that o be taken in the second quadrant, because only in this quadrant is sin $ positive and cos o negative for the same angle.

(3) Sin $ and cos o both negative. This requires that be taken in the third quadrant, because only in this quadrant are sin $ and cos o both negative for the same angle.

(4) Sin negative and cos • positive. This requires that o be taken in the fourth quadrant, because only in this quadrant is sin $ negative and cos o positive for the same angle.

Ex. 1. Solve the equations msin p= 332.76, and mcos & = 290.08, for m and $.

log m sin ø= 2.52213
log m cos p= 2.46252

log tan p= 0.05961 .:ø= 48° 55'.2.

log m sin ø= 2.52213
log sin ø= 9.87725

log m= 2.64488 .: m= 441.45. Ex. 2. Solve m sin ø=– 72.631, and m cos $ = 38.412.

Ans. ø= 117° 52'.3, m=- - 82.164. 84. Solve the equation a sin 0 + b cosø=c.

(1) a, b, and c being given, and $ required.

b Find in the tables the angle whose tangent is be B.

6 Then tan B, and (1) becomes

a(sin $ + tan ß cos ®)=c;
(sin $ cos ß + cos • sin ß

:C;
cos ß
sin(+ +B)= cos B=s

(2)

=

a

or

a

с

or

sin ß

a