Quadratic Trigonometric Equations
How do I solve quadratic trigonometric equations?
- A quadratic trigonometric equation is one that includes either , or
- Often the identity can be used to help solve the equation
- This can change an equation with both sine and cosine
- into an equation with only sine or cosine
- Solve the quadratic equation using any of the usual methods
- You may find it easier to rewrite it as an equation with a single letter
- e.g. writing as
- You may find it easier to rewrite it as an equation with a single letter
- A quadratic can give up to two solutions
- You must check whether solutions to the quadratic are valid solutions
- So and are the solutions of the quadratic
- Remember that solutions for and only exist for
- So may be a correct solution for the quadratic
- But it does not give a valid solution for the trigonometric equation!
- Solutions for exist for all values of
- You must check whether solutions to the quadratic are valid solutions
- After you solve the quadratic equation
- Find all solutions for the resulting trigonometric equation(s) within the given interval
- For the example above this would mean solving
- There will often be more than two trigonometric solutions for one quadratic equation
- Sketching a graph can help check how many solutions there should be in the given interval
- Find all solutions for the resulting trigonometric equation(s) within the given interval
Exam Tip
- Sketch the trig graphs on your exam paper
- Then you can refer back to them as many times as you need to
- Make sure you have found all of the solutions in the given interval
- And that you don't give solutions outside the interval
- For example if you get a negative solution but the interval is entirely positive
Worked example
Solve the equation , finding all solutions in the interval . Give your answers correct to 3 significant figures.
can be rearranged as
Substitute this to get the equation entirely in terms of
Expand the brackets and rearrange to get a quadratic equal to zero
This can be solved by factorising (it might help you to think of it as )
You could also solve it by using the quadratic formula
Or your calculator may be able to solve quadratics
has no solutions for because sine cannot be less than
So we only need to find solutions for
Start by finding the primary solution
The interval is given in radians, so we have to make sure the calculator is set up for radians!
Use symmetry properties of sine to find the secondary solution
Both those solutions are the interval , and there are no other solutions in the interval
(You could sketch the sine function to confirm that)