3.6.6 Quadratic Trigonometric Equations

Quadratic Trigonometric Equations

A quadratic trigonometric equation is one that includes either $\sin^{2} θ$ , $\cos^{2} θ$ or $\tan^{2} θ$
Often the identity $\sin^{2} θ + \cos^{2} θ = 1$ can be used to rearrange the equation into a form that is possible to solve
- If the equation involves both sine and cosine then the Pythagorean identity should be used to write the equation in terms of just one of these functions
Solve the quadratic equation using your GDC, the quadratic equation or factorisation
- This can be made easier by changing the function to a single letter
  - Such as changing ${2 \cos}^{2} θ - 3 \cos θ - 1 = 0$ to $2 c^{2} - 3 c - 1 = 0$
A quadratic can give up to two solutions
- You must consider both solutions to see whether a real value exists
- Remember that solutions for sin θ = k and cos θ = k only exist for -1 ≤ k ≤ 1
- Solutions for tan θ = k exist for all values of k
Find all solutions within the given interval
- There will often be more than two solutions for one quadratic equation
- The best way to check the number of solutions is to sketch the graph of the function

Sketch the trig graphs on your exam paper to refer back to as many times as you need to!
Be careful to make sure you have found all of the solutions in the given interval, being super-careful if you get a negative solution but have a positive interval

Solve the equation $11 \sin x - 7 = 5 \cos^{2} x$ , finding all solutions in the range $0 \leq x \leq 2 π$ .

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