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Trigonometric Equations, General Solution of some Standard Equations (Part 1) is considered one of the most asked concept.
51 Questions around this concept.
If
then the value of is :
In a , if , then the angle R is equal to
The general solution of the equation
is
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Let n be a fixed positive integer such that then
The number of solutions of the equation in the interval is:
Let is equal to
Let and is equal to:
The most general solution of is :
If , lies in the second quadrant. Then the value of is
has exactly 7 solutions in the interval , for the least value of n is Equal to.
Trigonometric Equations
Trigonometric equations are, as the name implies, equations that involve trigonometric functions.
Solution of Trigonometric Equation
The value of an unknown angle which satisfies the given trigonometric equation is called a solution or root of the equation. For example, 2sinӨ = 1, clearly Ө = 300 satisfies the equation; therefore, 300 is a solution of the equation. Now trigonometric equation ususally has infinite solutions due to periodic nature of trigonometric functions. So this equation also has (360+30)o,(720+30)o,(-360+30)o and so on, as its solutions.
Principal Solution
The solutions of a trigonometric equation that lie in the interval [0, 2π). For example, 2sinӨ = 1 , then the two values of sinӨ between 0 and 2π are π/6 and 5π/6. Thus, π/6 and 5π/6 are the principal solutions of equation 2sinӨ = 1.
General Solution
As trigonometric functions are periodic, solutions are repeated within each period, so, trigonometric equations may have an infinite number of solutions. The solution consisting of all possible solutions of a trigonometric equation is called its general solution.
Some Important General Solutions of Equations
General Solution of some Standard Equations (Part 1)
1. sin Ө = sin α
2. cos Ө = cos α
3. tan Ө = tan α
General Solution of some Standard Equations (Part 2)
4. sin2 Ө = sin2 α
Note:
The general solution of the equation cos2 Ө = cos2 α and tan2 Ө = tan2 α is also
Important Points to remember while solving trigonometric equations
While solving a trigonometric equation, squaring the equation at any step should be avoided as much as possible. If squaring is necessary, check the solution for values that do not satisfy the original equation.
Never cancel terms containing unknown terms on the two sides which are in product. It may cause the loss of a genuine solution.
The answer should not contain such values of angles which make any of the terms undefined or infinite.
Domain should not change while simplifying the equation. If it changes, necessary corrections must be made.
Check that the denominator is not zero at any stage while solving the equations.
Example:
Solve sin x + cos x = 1
Solution:
Hence only n = 0, 4, 8, 12, ..... and n = 1, 5, 9, .... satisfy the equation
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