Quadratic Trigonometric Equations

How are quadratic trigonometric equations solved?

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

Examiner Tip

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

Worked example

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

aa-sl-3-6-5-quadratic-trig-equations-we-solution

Quadratic Trigonometric Equations (DP IB Maths: AA HL)

Revision Note

How are quadratic trigonometric equations solved?

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1. Number & Algebra

1.1 Number & Algebra Toolkit

1.1.1 Standard Form

1.1.2 Laws of Indices

1.1.3 Partial Fractions

1.2 Exponentials & Logs

1.2.1 Introduction to Logarithms

1.2.2 Laws of Logarithms

1.2.3 Solving Exponential Equations

1.3 Sequences & Series

1.3.1 Language of Sequences & Series

1.3.2 Arithmetic Sequences & Series

1.3.3 Geometric Sequences & Series

1.3.4 Applications of Sequences & Series

1.3.5 Compound Interest & Depreciation

1.4 Simple Proof & Reasoning

1.4.1 Proof

1.5 Further Proof & Reasoning

1.5.1 Proof by Induction

1.5.2 Proof by Contradiction

1.6 Binomial Theorem

1.6.1 Binomial Theorem

1.6.2 Extension of The Binomial Theorem

1.7 Permutations & Combinations

1.7.1 Counting Principles

1.7.2 Permutations & Combinations

1.8 Complex Numbers

1.8.1 Intro to Complex Numbers

1.8.2 Modulus & Argument

1.8.3 Introduction to Argand Diagrams

1.9 Further Complex Numbers

1.9.1 Geometry of Complex Numbers

1.9.2 Forms of Complex Numbers

1.9.3 Complex Roots of Polynomials

1.9.4 De Moivre's Theorem

1.9.5 Roots of Complex Numbers

1.10 Systems of Linear Equations

1.10.1 Systems of Linear Equations

1.10.2 Algebraic Solutions

2. Functions

2.1 Linear Functions & Graphs

2.1.1 Equations of a Straight Line

2.2 Quadratic Functions & Graphs

2.2.1 Quadratic Functions

2.2.2 Factorising & Completing the Square

2.2.3 Solving Quadratics

2.2.4 Quadratic Inequalities

2.2.5 Discriminants

2.3 Functions Toolkit

2.3.1 Language of Functions

2.3.2 Composite & Inverse Functions

2.3.3 Symmetry of Functions

2.3.4 Graphing Functions

2.4 Other Functions & Graphs

2.4.1 Exponential & Logarithmic Functions

2.4.2 Solving Equations

2.4.3 Modelling with Functions

2.5 Reciprocal & Rational Functions

2.5.1 Reciprocal & Rational Functions

2.6 Transformations of Graphs

2.6.1 Translations of Graphs

2.6.2 Reflections of Graphs

2.6.3 Stretches Graphs

2.6.4 Composite Transformations of Graphs

2.7 Polynomial Functions

2.7.1 Factor & Remainder Theorem

2.7.2 Polynomial Division

2.7.3 Polynomial Functions

2.7.4 Roots of Polynomials

2.8 Inequalities

2.8.1 Solving Inequalities Graphically

2.8.2 Polynomial Inequalities

2.9 Further Functions & Graphs

2.9.1 Modulus Functions