Calculus I

Lesson 14: Solving Trigonometric Equations

In this lesson, we show how to solve equations for x that contain trig functions. Our general method will be to move all terms over to the left-hand side of the equation and find the roots of the resulting equation. We'll find these roots both analytically (by solving) and graphically by inspecting the plot and seeing where the curve crosses the x-axis.

Example 1

> restart:

> f1:= x -> sin(x) - sqrt(3)/2;

> plot(f1(x), x = 0..2*Pi, color = red, thickness=2);

> solve(f1(x) = 0, x);

Solutions are: and

Example 2

> f2:= x -> cos(x)* cos(x) - 1/2;

> plot(f2(x), x = 0..2*Pi, color = red, thickness=2);

> solve(f2(x) = 0, x);

Solutions are:

Example 3
sin(x) cos(x) = 0

> f3:= x -> sin(x) * cos(x) ;

> plot(f3(x), x = 0..2*Pi, color = red, thickness=2);

> solve(f3(x) = 0, x);

Solutions are:

Example 4

> f4:= x -> (tan(x) - 1) * ( 2 * sin(x) + 1);

> plot(f4(x), x = 0.. 2 * Pi, color = red, thickness=2);

> solve(f4(x) = 0, x );

> solve(tan(x) - 1 = 0, x);

> with(plots):

Warning, the name changecoords has been redefined

> a:= plot(2 * sin(x) + 1 , x = -Pi..2*Pi, color = red, thickness=2):

> b:= plot([-Pi/6,t,t=-1..1], color = blue):

> c:= plot([(11/6)*Pi,t, t = -1..1], color = blue):

> d:= plot([(7/6)*Pi,t ,t = -1..1], color = blue):

> display({a,b,c,d});

Solutions are:

Example 5

> f5:= x-> 2 * sin(x) * sin(x) - sin(x) - 1;

> plot(f5(x), x = 0..2*Pi, color = red, thickness=2);

> factor(2 * sin(x) * sin(x) - sin(x) - 1);

> solve(sin(x) -1 = 0,x);

Using problem 4 for 2 sin(x) + 1 = 0 we have solutions are: .

Example 6
sin(2x) + sin(x) = 0

> f6:= x -> sin (2*x) + sin(x);

> plot(f6(x), x = 0..2*Pi, color = red, thickness=2);

> simplify(sin (2*x) + sin(x));

We have: sin(x) ( 2 cos(x) + 1 ) = 0, i.e.,

sin(x) = 0 OR 2 cos(x) + 1 = 0.

> plot(2*cos(x) + 1, x = 0..2*Pi, color = red, thickness=2);

> solve(2*cos(x) + 1 = 0, x);

Solutions are:

Example 7
cos(x) + cos(2x) = 0

> f7:= x ->cos(x) + cos(2*x);

> plot(f7(x), x = 0..2*Pi, color = red, thickness=2);

> simplify( cos(x) + cos(2*x));

> factor(%);

We obtain solutions from:

cos(x) = -1 OR 2 cos(x) = 1.

> plot(cos(x) + 1, x = 0..2*Pi, color = red, thickness=2);

> solve(cos(x) + 1 = 0,x);

> plot(2 * cos(x) - 1, x = 0..2 * Pi, color = red, thickness=2);

> solve(2 * cos(x) - 1,x);

Solutions are:

Example 8

2 tan(x) sin(x) - tan(x) = 0

> f8:= x -> 2 * tan(x) * sin(x) - tan(x);

> plot(f8(x), x = 0..2*Pi, color = red, thickness=2);

> factor(2 * tan(x) * sin(x) - tan(x));

Solutions are obtained from

tan(x) = 0 OR sin(x) =

> plot(sin(x) - 1/2, x = 0..Pi, color = red, thickness=2);

> solve(sin(x) - 1/2 = 0, x);

Solutions are:

Example 9
2 cos(x) + sec(x) = 3

> f9:= x-> 2 * cos(x) + sec(x) - 3;

> plot(f9(x), x = 0..2*Pi, color = red, thickness=2);

> simplify(2 * cos(x) + sec(x) - 3);

> factor(2 * cos(x) * cos(x) - 3* cos(x) + 1);

Solutions are obtained from:

cos(x) = (1/2 OR cos(x) = 1.

> solve(cos(x) - 1/2 = 0,x);

we have:

> f9((1/3)*Pi);

> f9((5/1)*Pi);

> f9(0);

> f9(2 *Pi);

Solutions are: .

Example 10
2 sin(x) + csc(x) = 3

> f10:= x -> 2 * sin(x) + csc(x) - 3;

> plot(f10(x), x = 0..Pi/2, color = red, thickness=2);

> factor(2 * sin(x) * sin(x) + 1 - 3* sin(x));

Solutions are obtained from:

sin(x) = 1/2 OR sin(x) = 1.

we obtain:

> f10((1/6)*Pi);

> f10((11/6)*Pi);

> f10((1/2)*Pi);

Solutions are: .

Example 11
sin(x) + 1 = cos(x)

> f11:= x -> sin(x) + 1 - cos(x);

> plot(f11(x), x = 0..2*Pi, color = red, thickness=2);

take the equation sin(x) + 1 = cos(x) and square both sides to get:

sin(x)^2 + 2 sin(x) + 1 = cos(x)^2 = 1 - sin(x)^2

OR

2 sin(x)^2 + 2 sin(x) = 0

OR

2 sin(x) ( sin(x) + 1) = 0.

> f11(0);

> f11(Pi);

> f11(2 * Pi);

> f11((3/2) * Pi);

Solutions are: .