3.6.5 Linear Trigonometric Equations

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
- The graphs of trigonometric functions

If you are using your GDC it will only give you the principal value and you need to find all other solutions for the given interval
Sketch out the CAST diagram and the trig graphs on your exam paper to refer back to as many times as you need to

Solve the equation $2 \cos x = - 1$ , finding all solutions in the range $- π \leq x \leq π$ .

aa-sl-3-6-4-trig-equations-sinx--k-we-solution

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

If you transform the interval, remember to convert the found angles back to the final values at the end!
If you are using your GDC it will only give you the principal value and you need to find all other solutions for the given interval
Sketch out the CAST diagram and the trig graphs on your exam paper to refer back to as many times as you need to

Solve the equation $2 \cos (2 x - 30 °) = - 1$ , finding all solutions in the range $- 360 ° \leq x \leq 360 ° .$

aa-sl-3-6-4-trig-equations-sinaxb--k-we-solution

DP IB Maths: AA HL

3.6.5 Linear Trigonometric Equations