Question

If tan A= 1/3, tan B=1/4 where 0<B<31/2 then sin (A+B) is​

Answer 1

Answer:TRIGONOMETRIC EQUATIONS

An equation involving one or more trigonometrical ratios of unknown angle is called a trigonometric equation e.g.  cos2  x – 4 sin x = 1

It is to be noted that a trigonometrical identity is satisfied for every value of the unknown angle where as trigonometric equation is satisfied only for some values (finite or infinite) of unknown angle.

e.g. sec2 x – tan2 x = 1 is a trigonometrical identity as it is satisfied for every value of x Î R.

SOLUTION OF A TRIGONOMETRIC EQUATION

A value of the unknown angle which satisfies the given equation is called a solution of the   equation e.g.  sin q = ½  Þq = p/6 .

General   Solution

Since trigonometrical functions are periodic functions, solutions of trigonometric equations can be generalized with the help of the periodicity of the trigonometrical functions. The solution consisting of all possible solutions of a trigonometric equation is called its general solution.

We use the following formulae for solving the trigonometric equations:

·         sin q  = 0 Þ  q = np,

·         cos q = 0 Þq = (2n + 1),

·         tan q = 0  Þ  q =  np,

·         sin q = sin a  Þq =  np + (–1)na,         where aÎ [–p/2,  p/2]

·         cos q  = cos aÞq = 2np  ±  a, where aÎ [ 0, p]

·         tan q = tan a  Þ  q = np + a,                 where aÎ ( –p/2, p/2)

·         sin2 q = sin2 a , cos2 q = cos2 a,  tan2q = tan2 aÞq = np±a,

·         sin q = 1 Þq = (4n + 1),

·         cos q = 1 Þ  q = 2np ,

·         cos q = –1 Þ  q = (2n + 1)p,

·         sin q =  sin a  and  cos q =  cos aÞ  q = 2np + a.

Note:

·         Everywhere in this chapter n is taken as an integer, If not stated otherwise.

·         The general solution should be given unless the solution is required in a specified interval.

·         a is taken as the principal value of the angle. Numerically least angle is called the principal value.

Method for finding principal value

Suppose we have to find the principal value of  satisfying the equation sin = – .

Since sin is negative,  will be in 3rd or 4th quadrant. We can approach 3rd or 4th quadrant from two directions. If we take anticlockwise direction the numerical value of the angle will be greater than . If we approach it in clockwise direction the angle will be numerically less than . For principal value, we have to take numerically smallest angle.

So for principal value :

1.         If the angle is in 1 st or 2nd  quadrant we must select anticlockwise direction and if the angle if the angle is in 3rd or 4th quadrant, we must select clockwise direction.

2.         Principal value is never numerically greater than .

3.         Principal value always lies in the first circle (i.e. in first rotation)

On the above criteria  will be  or . Among these two  has the least numerical value. Hence  is the principal value of  satisfying the equation sin = –.

Algorithm to find the principle argument:

Step 1:           First draw a trigonometric circle and mark the quadrant, in which the angle may lie.

Step 2:           Select anticlockwise direction for 1st and 2nd quadrants and select clockwise direction for 3rd  and 4th quadrants.

Step 3:           Find the angle in the first rotation.

Step 4:           Select the numerically least angle among these two values. The angle thus found will be the principal value.

Step 5:           In case, two angles one with positive sign and the other with negative sign qualify for the numerically least angle, then it is the convention to select the angle with positive sign as principal value.

Step-by-step explanation: