Co-terminal Angles

The angles (2p + A), (4p + A)......have the same initial and terminal arm as the angle A and so these angles are called co-terminal angles. For all such angles, the value of any trigonometric ratio is the same. Thus, all co-terminal angles are trigonometrically equivalent.

Let us consider the simplest form of a trigonometric equation, where a trigonometric function is equal to a constant. For example

Since the trigonometric functions of all co-terminal angles are same, and since an angle may have infinite number of co-terminals, therefore the equation  can have infinite number of roots.

For instance,

All these solutions put together in a compact form or forms are called complete solution or general solution of the equation.

To find the general solution of the following equations

(i) sin q = 0 (ii) tan q = 0 (iii) cos q = 0 (iv) cot q = 0.

Suggested answer:

General solution of (i) sin q = 0 (ii) tan q = 0

From the figure, we have

This is possible only if the terminal side of the angle q falls in line with OX or OX'.

i.e., when q = 0, p, 2p, ........np, where n Z.

i.e., 0 = np

\ q = np is a general solution of the equation sin q = 0.

values of q, for which y = 0. Hence the general solution of tan q = 0 is

q = np.

iii) General solution of cos q = 0

cos q = 0 only when x= 0 i.e., the terminal side of q falls in line with OY or OY ' (or coincides with OY or OY ') i.e., when

iv) cot q = 0

From the figure, we have

cot q = 0 only when x = 0.

When the terminal side of q falls in line with OX or OX' (coincides with OX or OX'), i.e., when

Note:

The equation sec q = 0, cosec q = 0 can have no solutions as the values of sec q and cosec q cannot be numerically less than 1.