TRIGONOMETRIC EQUATION

GENERAL SOLUTION OF TRIGONOMETRIC EQUATION

Trig. Function are periodic, there are many be an infinite numbers in a general solution. So general solutions are usually given expressions in terms of n, where n is an integer.

 

General Solution for angles in radian

 

Solving Trigonometric Equation

Type 1.            Equation in which one function of a single is involved

 

Example

Solution

Example

Find all the x which satisfy 3sin x = 4cos x

Solution

Example

Find the general value of

Solution

Example

Solution

Types 2

Equation expressible in terms of one trig. Ratio of the unknown angle.

This method could be helpful

1.                  Square relation

2.                  Double angle formula

Example

Find all the values of f  in the range satisfying the equation

Solution

 

 

Example

Find the general solution  for which

 

Solution

 

LINEAR COMBINATION OF COS AND SIN

 is called a linear combination of cos and sin. It can be expressed as a single term in the form:

Where R > 0 and a is acute angle. R and a depend on a and b. To find the value their values, use a compound angle formula to expand the chosen single term first. Then compare coefficient  of cos q  and sinq  the two equivalent expression to obtain a = …..

and b = ….. Squaring these quantities and adding gives R² and hence R. Dividing them gives tan a an hence a

 

 

To find the value of R and a when

 acos q + bsin q =Rcos(q - a ), and R is positive with

 

 By using similar method the values of R and a can be determined in the following cases

In all cases R =

 

Solving equation of the form a cos q + b sin q =

Method (1)

 

Method2

Example

Given that  where R and a  are independent of q and R  is positive, Hence or otherwise, find the value of R and a Hence or otherwise , find the value of  q between – 180^o and 180^o  which satisfy the equation

 

Solution

  

 

Example

Prove that

 

Solution