Solving Trigonometric Equations (Cambridge (CIE) IGCSE Additional Maths): Revision Note

Linear Trig Equations

How are trigonometric equations solved?

• Trigonometric equations can have an infinite number of solutions

  • For an equation in sin or cos you can add 360° or 2π to each solution to find more solutions

  • For an equation in tan you can add 180° or π to each solution

• When solving a trigonometric equation you will be given a range of values within which you should find all the values

• Solving the equation normally and using the inverse function on your calculator or your knowledge of exact values will give you the primary value

• The secondary values can be found with the help of:

  • The unit circle

    • By using the CAST diagram which shows where each function has positive solutions

  • The graphs of trigonometric functions

    • By sketching the graph (see Graphs of Trigonometric Functions) you can read off all the solutions in a given range (or interval)

• By using trigonometric identities you can simplify harder equations

• You may be asked to use degrees or radians to solve trigonometric equations

  • Make sure your calculator is in the correct mode

  • Remember common angles

    • 90° is ½π radians

    • 180° is π radians

    • 270° is 3π/2 radians

    • 360° is 2π radians 

How are trigonometric equations of the form sin x = k solved?

• It is a good idea to sketch the graph of the trigonometric function first

  • Use the given range of values as the domain for your graph

  • The intersections of the graph of the function and the line y = k will show you

    • The location of the solutions

    • The number of solutions

  • You will be able to use the symmetry properties of the graph to find all secondary values within the given range of values

• The method for finding secondary values are:

  • For the equation sin x = k the primary value is x₁ = sin ^-1 k

    • A secondary value is x₂ = 180° - sin ^-1 k

    • Then all values within the range can be found using x₁ ± 360n and x₂ ± 360n where n ∈  

  • For the equation cos x = k the primary value is x₁ = cos ^-1 k 

    • A secondary value is x₂ = - cos ^-1 k

    • Then all values within the range can be found using x₁ ± 360n and x₂ ± 360n where n ∈  

  • For the equation tan x = k  the primary value is x = tan ^-1 k

    • All secondary values within the range can be found using x ± 180n where n ∈   

How do I use the CAST diagram?

What about more complicated trig equations? 

• Some trig equations could involve a function of x or θ (see Transformations of Trigonometric Functions)

• Trigonometric equations in the form sin(ax + b) can be solved in more than one way

• The easiest method is to consider the transformation of the angle as a substitution

  • For example let u = ax + b

• Transform the given interval for the solutions in the same way as the angle

  • For example if the given interval is 0° ≤ x ≤ 360° the new interval will be

  • (a (0°) + b) ≤ u ≤ (a (360°) + b)

• Solve the function to find the primary value for u

• Use either the unit circle or sketch the graph to find all the other solutions in the range for u

• Undo the substitution to convert all of the solutions back into the corresponding solutions for x

• Another method would be to sketch the transformation of the function

  • If you use this method then you will not need to use a substitution for the range of values

What about equations using sec, cosec, and cot?

• Equations in the form (or  or ) are not solvable using your calculator, as it does not have an inverse function for these

• Use the definitions of these functions to rewrite the equation in terms of , , or 

  • and then solve as normal, remembering to find all the solutions within the given domain of

• It is often useful to swap sec, cosec, and cot for their definitions as one of the first steps when solving an equation

Quadratic Trig Equations

How are quadratic trigonometric equations solved?

• A quadratic trigonometric equation is one that includes either , or 

• Often the identity 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

  • If the equation involves a mixture of regular trig functions (sin, cos, and tan) as well as reciprocal functions (sec, cosec, and cot) it may be useful to:

    • rewrite the reciprocal trig functions using their definitions,

      • use the identities and 

• Solve the quadratic equation using the quadratic equation or factorisation

  • This can be made easier by changing the function to a single letter

    • Such as changing  to 

• 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

• You may be asked to use degrees or radians to solve trigonometric equations

  • Make sure your calculator is in the correct mode

  • Remember common angles

    • 90° is ½π radians

    • 180° is π radians

    • 270° is 3π/2 radians

    • 360° is 2π radians

Strategy for Further Trig Equations

How to approach solving trig equations

• You can solve trig equations in a variety of different ways

  • Sketching a graph (see Graphs of Trig Functions)

  • Using trigonometric identities (see Simple Trig Identities and Further Trig identities)

  • Using the CAST diagram (see Linear Trig Equations)

  • Factorising quadratic trig equations (see Quadratic Trig Equations)

• You may be asked to use degrees or radians to solve trigonometric equations

  • Make sure your calculator is in the correct mode

  • Remember common angles

    • 90° is ½π radians

    • 180° is π radians

    • 270° is 3π/2 radians

    • 360° is 2π radians 

• If you’re having trouble solving a trig equation, this flowchart might help: