MHB Show the trigonometric equation has no solutions

Click For Summary
The trigonometric equation $\sin (\cos a) = \cos (\sin a)$ is examined for potential solutions. It is established that the range of $\sin(\cos a)$ is limited to the interval [-1, 1], while $\cos(\sin a)$ also falls within the same range. However, through analysis, it is demonstrated that these two functions do not intersect at any point within their respective ranges. Consequently, it is concluded that the equation has no solutions. The proof relies on the properties and behaviors of the sine and cosine functions.
anemone
Gold Member
MHB
POTW Director
Messages
3,851
Reaction score
115
Show that the trigonometric equation $\sin (\cos a)= \cos (\sin a)$ has no solutions.
 
Mathematics news on Phys.org
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
 

Thread 'Area of Overlapping Squares'

Here is a little puzzle from the book 100 Geometric Games by Pierre Berloquin. The side of a small square is one meter long and the side of a larger square one and a half meters long. One vertex of the large square is at the center of the small square. The side of the large square cuts two sides of the small square into one- third parts and two-thirds parts. What is the area where the squares overlap?
View full post »

Similar threads

High School Trigonometric Identity involving sin()+cos()
  • · Replies 11 ·
Replies
11
Views
2K
High School Can 2(sinx) be called a 'trigonometric function'?
  • · Replies 28 ·
Replies
28
Views
3K
High School What values of m as a function of q satisfy this trigonometric equation?
  • · Replies 5 ·
Replies
5
Views
1K
MHB Solve Situational Problems Involving Trigonometric Identities
  • · Replies 3 ·
Replies
3
Views
2K
MHB Unraveling a Trigonometric Mystery
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Question ,trigonometric identities equation and functions ?
  • · Replies 4 ·
Replies
4
Views
2K
MHB Is this Trigonometric Expression a Constant Function of x?
  • · Replies 2 ·
Replies
2
Views
1K
MHB How to Solve the Trigonometric Equation 6*Sin^2(x) - 3*Sin^2(2x) + Cos^2(x) = 0?
  • · Replies 6 ·
Replies
6
Views
3K
MHB What Is the Value of $\tan^2 a + 2\tan^2 b$ Given the Trigonometric Condition?
  • · Replies 5 ·
Replies
5
Views
2K
MHB Finding Points of Intersection for Trigonometric Functions
  • · Replies 8 ·
Replies
8
Views
3K