Show the trigonometric equation has no solutions

In summary, a trigonometric equation having no solutions means that there are no possible values for the variable to make the equation true. To show this, one can use algebraic manipulation and trigonometric identities to arrive at a contradiction. There are two special cases when a trigonometric equation has no solutions, and it is possible for an equation with multiple trigonometric functions to have no solutions. It is important to show that a trigonometric equation has no solutions in order to determine its validity and gain a better understanding of trigonometric functions.
  • #1
anemone
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Show that the trigonometric equation $\sin (\cos a)= \cos (\sin a)$ has no solutions.
 
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  • #2
anemone said:
Show that the trigonometric equation $\sin (\cos a)= \cos (\sin a)$ has no solutions.

if sin (x) = cos (y) then

$x+y = \pi/2 +2npi$ or $(2n+1)pi-\pi/2$

so lowest | cos a + sin a | = $\pi/2$

so |cos a + sin a| =$ \pi/2$ should have a solution

but |cos a + sin a | <= $\sqrt(2)$

so no solution
 

FAQ: Show the trigonometric equation has no solutions

What does it mean for a trigonometric equation to have no solutions?

When a trigonometric equation has no solutions, it means that there are no possible values for the variable that satisfy the equation. In other words, there is no angle or set of angles that can be substituted into the equation to make it true.

How can you show that a trigonometric equation has no solutions?

To show that a trigonometric equation has no solutions, you can use algebraic manipulation and trigonometric identities to arrive at a contradiction. This means that you will end up with an equation that is always false, no matter what value is substituted for the variable.

Are there any special cases when a trigonometric equation has no solutions?

Yes, there are two special cases when a trigonometric equation has no solutions. The first is when the equation contains a tangent or cotangent function and the angle being solved for is equal to a multiple of 90 degrees. The second is when the equation contains a secant or cosecant function and the angle being solved for is equal to a multiple of 180 degrees.

Can a trigonometric equation have no solutions if it contains more than one trigonometric function?

Yes, a trigonometric equation can have no solutions if it contains more than one trigonometric function. In this case, you will need to use multiple trigonometric identities and algebraic manipulation to arrive at a contradiction.

Why is it important to show that a trigonometric equation has no solutions?

Showing that a trigonometric equation has no solutions is important because it allows us to determine if the equation is valid or not. If an equation has no solutions, then it is not a valid mathematical statement and any conclusions drawn from it would be incorrect. It also helps us to better understand the properties and behavior of trigonometric functions.

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