How Many Solutions Does the Trigonometric Equation Have in the Given Interval?

In summary: Here are some examples:-When ##x## is a complex number.-When ##x## is a parameter.-When ##x## is a negative number.-When ##x## is not a real number.
  • #1
Vavi Ask
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Homework Statement


The number of solutions of the equation sin2x –2cosx + 4sinx = 4 in the interval [0, 5π] is what?

Homework Equations


sin2x=2sinxcosx

The Attempt at a Solution


sin2x-2cosx+4sinx=4
⇒2sinxcosx-2cosx+4sinx=4
⇒sinxcosx-cosx+2sinx=2
⇒cosx (sinx-1)=2-2sinx
⇒cosx (sinx-1)=2 (1-sinx)
⇒cosx (sinx-1)= -2 (sinx-1)
⇒cosx = -2
∴ There is no solution
 
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  • #2
Your last step assumes something that is not necessarily true.
 
  • #3
WolframAlpha finds at least one solution in [0,π]...
 
  • #4
Do you have the ability to plot the expression on the left side (with a calculator or spreadsheet)? Then see if it crosses 4 anywhere. That is a start to see how many solutions you are dealing with.

Like @Orudruin indicated - you divided both sides by an expression. There is a time when you don't want to do that.
 
  • #5
scottdave said:
Do you have the ability to plot the expression on the left side (with a calculator or spreadsheet)? Then see if it crosses 4 anywhere. That is a start to see how many solutions you are dealing with.

I found indeed several with my calculator but solving it algebraically

After the step cosx (sinx-1)=2 (1-sinx) as the OP did i got lost. I am strying to solve it as well without solving it for the OP. But why you can't divide after this step, what is wrong about that?
 
  • #6
ElectricRay said:
I found indeed several with my calculator but solving it algebraically

After the step cosx (sinx-1)=2 (1-sinx) as the OP did i got lost. I am strying to solve it as well without solving it for the OP. But why you can't divide after this step, what is wrong about that?
First of all, be aware that posting complete solutions before the OP has solved the problem is not allowed. See the forum rules.

Regarding your question: In which cases is it not ok to divide by a real number ##x##?
 
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Related to How Many Solutions Does the Trigonometric Equation Have in the Given Interval?

What is Trigonometric Solutions?

Trigonometric Solutions refer to the methods and techniques used to solve problems involving triangles and trigonometric functions, such as sine, cosine, and tangent.

What are the three basic trigonometric functions?

The three basic trigonometric functions are sine, cosine, and tangent. These functions are used to find the ratios of sides in a right triangle.

How are trigonometric solutions used in real life?

Trigonometric solutions are used in various fields such as engineering, physics, and navigation. They are also used in everyday tasks such as measuring heights and distances.

What are the common applications of trigonometric solutions in mathematics?

Trigonometric solutions are commonly used in solving problems involving triangles, such as finding missing angles or sides, as well as in calculus and other advanced mathematical concepts.

What is the unit circle and how is it related to trigonometric solutions?

The unit circle is a circle with a radius of 1 centered at the origin on a coordinate plane. It is used to visualize and understand the values of trigonometric functions. The coordinates of points on the unit circle correspond to the sine and cosine values of the corresponding angle.

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