Trigonometric Functions-II, Trigonometric Functions of Submultiples of Angles, Trigonometric Equations, to Find the General Solution of the Equation sinθ = sinα

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Trigonometric Functions of Submultiples of Angles:

are called submultiples of .

It has been proved that

, ,

Replacing by , we easily get the following formulae for the sub-multiple:

, ,

We will choose either the positive or the negative sign depending on whether corresponding value of the function is positive or negative for the value of .

This will be clear from the following examples

Example:

Find the values of and .

Solution:

We use the formula and take the lower sign, i.e., negative sign, because is negative.

Similarly,

Trigonometric Equations:

  • You are familiar with the equations like simple linear equations, quadratic equations in algebra.

  • You have also learnt how to solve the same.

  • Thus, (i) gives one value of as a solution.

(ii) gives two values of .

  • You must have noticed, the number of values depends upon the degree of the equation.

  • Now we need to consider as to what will happen in case and are replaced by trigonometric functions.

  • Thus solution of the equation , will give

and

  • Clearly, the solution of simple equations with only finite number of values does not necessarily hold good in case of trigonometric equations.

  • So, we will try to find the ways of finding solutions of such equations.

To Find the General Solution of the Equation Sin Θ=Sin Α

It is given that ,

Or

Either or

or ,

or ……..

From, we get

as the general solution of the equation

To Find the General Solution of the Equation Cos Θ=Cos Α

It is given that, ,

Either, or

or ,

or ……..

From, we get

as the general solution of the equation

To Find the General Solution of the Equation Tan Θ=Tan Α

It is given that, ,

Similarly, for , the general solution is

And, for , the general solution is

And, for , the general solution is

Example:

Find the general solution of the following equations:

(i)

(ii)

(iii)

Solution:

(i)

(ii)

(iii)

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