5. Solving Trigonometric Equations
by M. Bourne
Trigonometric equations can be solved using the algebraic methods and trigonometric identities and values discussed in earlier sections. You may wish to go back and have a look at Trigonometric Functions of Any Angle, where we see the background to the following solutions.
Need Graph Paper?
A painless way to solve these is using a graph. Where the graph cuts the x-axis, there are your solutions. Graphs also help you to understand why sometimes there is one answer, and sometimes many answers. I use Scientific Notebook or LiveMath to graph the functions for me.
Example 1
Solve the equation 2 cos θ − 1 = 0 for 0 ≤ θ < 2π.
Example 2
Revision Tip
Getting lost in this section? See the background at Trigonometric Functions of Any Angle
Graphically solve the equation 2 cos2x − sin x − 1 = 0 such that 0 ≤ θ < 2π.
LiveMath answer:
Solving Equations Involving Multiples of θ
Example 3
Solve the equation sin 2θ = 0.8 for 0 ≤ θ < 2π.
Example 4
Solve the equation 
for
0 ≤ θ < 2π.
Example 5
Solve the equation 6 sin2θ − sin θ − 1 = 0 for 0 ≤ θ < 2π.
Example 6
Solve the equation 
for
0 ≤ θ < 2π.
Example 7
Solve the equation tan 2θ − cot 2θ = 0 for 0 ≤ θ < 2π.
Exercises
Note 1: "Analytically" means use the methods and formulas from previous sections. It means don't just use a graph to solve it.
Note 2: However, I always use a graph to check my analytical work. I can see immediately if some error has occurred. I encourage you to do the same!
1. Solve the trigonometric equation analytically
4 tan x − sec2 x = 0 (for 0 ≤ x < 2π)
2. Solve the trigonometric equation analytically for 0 ≤ x < 2π:
sin 2x cos x − cos 2x sin x = 0
Need Graph Paper?
3. Solve the given trigonometric equation analytically and by graphical method (for 0 ≤ x < 2π):
sin 4x − cos 2x = 0
Didn't find what you are looking for on this page? Try search:
The IntMath Newsletter
Sign up for the free IntMath Newsletter. Get math study tips, information, news and updates each fortnight. Join thousands of satisfied students, teachers and parents!
Trigonometry Lessons on DVD
Easy to understand trigonometry lessons on DVD. See samples before you commit.
More info: Trigonometry videos
Book mark this page
Add this page to Del.icio.us, Furl, Digg, StumbleUpon, Google, whatever...
Need a break? Play a math game. Well, they all involve math... No, really!
