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.

This is a trigonometric equation. To solve it we must

1

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

2

12

=

2

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

.. 0 30° is a solution of the equation.

Ꮎ ;

.. 0 = n + ( − 1)"

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

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

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

Therefore one value of 0 which satisfies the equation is 60°.

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

Find a value of which will satisfy the following equations:

4. cos 0= cos 2 0.

5. 2 cos 0=

= sec 0.

Ans. π.

45°.

where a and b are given, and the values of m and are required.

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

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 o has the sign of a, and cos the sign of b, and hence & must be taken in the quadrant denoted by these signs. These cases may be considered as follows:

(1) Sin and cos & 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 & negative. This requires that be taken in the second quadrant, because only in this quadrant is sin positive and cos & negative for the same angle. (3) Sin and cos & both negative. This requires that be taken in the third quadrant, because only in this quadrant are sin and cos & both negative for the same angle. (4) Sin negative and cos & positive. This requires that 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 = 332.76, and m cos = 290.08, for m and 4.

a sin +b cos &= c.

a, b, and c being given, and required.

Find in the tables the angle whose tangent is

82.164.

tan ẞ cos 4) = c;