Trigonometric Equations

An equation which contains Trigonometric Functions is known as trigonometric equations.
For example: Cos p = ½.
When we solve Trigonometric Equations using different Relations then the equation is converted to form through which value of other variable can be obtained. The roots of trigonometric function are obtained by the Inverse Trigonometric Functions.
Suppose we have sin p + sin 2p + sin 3p = 0;
This trigonometric equation can be reduced to form 2 sin 2p cos p + sin 2p = 0;
Or we can also write as:
Sin 2p (2 cos p + 1) = 0;
When we solve this equation we get the value of sin 2p as 0;
Sin 2p (2 cos p + 1) = 0;
Sin 2p = 0;
And the value of cos is:
2 cos p + 1 = 0;
2 cos p = -1;
Cos p = -1/2;
So the value of cos p is -1/2.
This equation result gives the result of the trigonometric equations.
So p = ½ arcsin 0 = n ⊼/2;
P = arcos (-1/2);
P = 2/3 ⊼ (3n + ½);
Where, the value of ‘n’ is either positive or negative Integer.
Now we will see how to solve the trigonometric equations.
Suppose we have the trigonometric equation 4 tan³ P – tan P = 0, that lies in the interval [0, 2⊼]. Then it can be solved as shown below:
So the given trigonometric equation is: 4 tan³ P – tan P = 0;
We can write the equation as:
=> 4 tan³ P – tan P = 0;
=> Tan p (4 tan² P – 1) = 0;
So the value of tan ‘P’ is 0;
Or tan P = + 1/√3. For every value of P ∈ [0, 2⊼],
Tan P = 0;
It means the value of ‘P’ is 0, ⊼ or 2⊼.
While
Tan P = 1/√3,
P = ⊼ / 6 or 7⊼/ 6,
Tan P = -1/√3,
P = 5⊼ / 6 or 11⊼/ 6.