To solve trigonometric equations, several identities and formulas are used i.e:
For basic trigonometric equations, we follow the following steps to solve them:
1. Make sine, cosine or tangent the subject.
2. Use any method including a calculator to find basic angles.
3. Using quadrants, find all solutions in the given range.
Q. Solve the equation for the range
Using a calculator we can find that the basic angle
Now we know that cos is positive in the first quadrant and in the fourth quadrant hence another value of would be:
Refer to Fig 1.
We know that:
Dividing 1 by 2 we get:
This is called an ”Identity”.
Using pythagoras theorem:
Dividing the whole equation by :
Substituting the values:
We get another identity:
where means ”equivalent to” or ”identical to”.
These identities are very useful when solving many trigonometric equations.
Q. Prove the identity
Taking left hand side only:
Similarly, there are a few other formulas that help solve trigonometric equation: