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Find All Solutions of the Equation Sin X 1 0

L14-trigEquations.mws


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
sin(x) = sqrt(3)/2

> restart:

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

f1 := proc (x) options operator, arrow; sin(x)-1/2*...

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

[Maple Plot]

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

1/3*Pi

Solutions are: Pi/3 and 2/3*Pi

Example 2
cos(x)^2 = 1/2

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

f2 := proc (x) options operator, arrow; cos(x)^2-1/...

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

[Maple Plot]

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

1/4*Pi, 3/4*Pi

Solutions are: 1/4*Pi, 3/4*Pi, 5/4*Pi, 7/4*Pi

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

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

f3 := proc (x) options operator, arrow; sin(x)*cos(...

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

[Maple Plot]

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

1/2*Pi, 0

Solutions are: 0, 1/2*Pi, Pi, 3/2*Pi, 2*Pi

Example 4
(tan(x)-1)*(2*sin(x)+1) = 0

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

f4 := proc (x) options operator, arrow; (tan(x)-1)*...

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

[Maple Plot]

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

1/4*Pi, -1/6*Pi, -5/6*Pi

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

1/4*Pi

> 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});

[Maple Plot]

Solutions are: 1*Pi/4, 5*Pi/4, 11*Pi/6, 7*Pi/6

Example 5
2*sin(x)^2-sin(x)-1 = 0

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

f5 := proc (x) options operator, arrow; 2*sin(x)^2-...

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

[Maple Plot]

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

(2*sin(x)+1)*(sin(x)-1)

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

1/2*Pi

Using problem 4 for 2 sin(x) + 1 = 0 we have solutions are: 11/6*Pi, 7/6*Pi and 1/2*Pi .

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

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

f6 := proc (x) options operator, arrow; sin(2*x)+si...

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

[Maple Plot]

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

2*sin(x)*cos(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);

[Maple Plot]

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

2/3*Pi

Solutions are: 0, Pi, 2*Pi, 2/3*Pi, 4/3*Pi

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

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

f7 := proc (x) options operator, arrow; cos(x)+cos(...

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

[Maple Plot]

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

cos(x)+2*cos(x)^2-1

> factor(%);

(cos(x)+1)*(2*cos(x)-1)

We obtain solutions from:

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

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

[Maple Plot]

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

Pi

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

[Maple Plot]

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

1/3*Pi

Solutions are: Pi, 1/3*Pi, 5/3*Pi

Example 8

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

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

f8 := proc (x) options operator, arrow; 2*tan(x)*si...

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

[Maple Plot]

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

tan(x)*(2*sin(x)-1)

Solutions are obtained from

tan(x) = 0 OR sin(x) = 1/2

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

[Maple Plot]

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

1/6*Pi

Solutions are: 0, Pi, 2*Pi, 1/6*Pi, 5/6*Pi

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

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

f9 := proc (x) options operator, arrow; 2*cos(x)+se...

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

[Maple Plot]

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

(2*cos(x)^2+1-3*cos(x))/cos(x)

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

(2*cos(x)-1)*(cos(x)-1)

Solutions are obtained from:

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

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

1/3*Pi

we have: 1/3*Pi, 5/3*Pi, 0, 2*Pi

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

0

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

-6

> f9(0);

0

> f9(2 *Pi);

0

Solutions are: 1/3*Pi, 0, 2*Pi .

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

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

f10 := proc (x) options operator, arrow; 2*sin(x)+c...

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

[Maple Plot]

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

(2*sin(x)-1)*(sin(x)-1)

Solutions are obtained from:

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

we obtain: 1/6*Pi, 5/6*Pi, 1/2*Pi

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

0

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

-6

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

0

Solutions are: 1/6*Pi and 1/2*Pi .

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

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

f11 := proc (x) options operator, arrow; sin(x)+1-c...

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

[Maple Plot]

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);

0

> f11(Pi);

2

> f11(2 * Pi);

0

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

0

Solutions are: 0, 2*Pi, 3/2*Pi .

Find All Solutions of the Equation Sin X 1 0

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