The sum of the first eight terms of the sequence {Inx, Inx^2y, Inx^3y^2,...} is given by 4(aInx+bIny). Find a and b.
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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.
Example 1: Iftan = – 1, then will lie in 2nd or 4th quadrant.
For 2nd quadrant we will select anticlockwise and for 4th quadrant. we will select clockwise direction.
In the first circle two values and are obtained.
Among these two, is numerically least angle. Hence principal value is .
Example 2: If cos = , then will lie in 1st or 4th quadrant.
For 1st quadrant, we will select anticlockwise direction and for 4th quadrant, we will select clockwise direction.
In the first circle two values and are thus found.
Both and –