Solving Trigonometric Equations (CIE IGCSE Additional Maths)

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

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

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: