Solve Trig Eqn: Find & Combine 2 Solutions for x

Click For Summary
SUMMARY

The discussion focuses on solving the trigonometric equation \(\cos x + \sin x = 0\). Two expressions for the solutions were presented: \(x = n\pi - \frac{\pi}{4}\) and \(x = \frac{(n\pi)}{2} + \frac{\pi}{4}\). It was confirmed that both expressions are equivalent when \(n\) is defined correctly, specifically as any odd integer for the second expression. The clarification ensures that both forms represent the same set of solutions.

PREREQUISITES
  • Understanding of trigonometric functions and their properties
  • Familiarity with solving trigonometric equations
  • Knowledge of integer sequences and their implications in equations
  • Basic algebraic manipulation skills
NEXT STEPS
  • Study the unit circle and its relation to trigonometric functions
  • Learn about the periodicity of trigonometric functions
  • Explore the concept of odd and even integers in mathematical expressions
  • Investigate other methods for solving trigonometric equations
USEFUL FOR

Students of mathematics, educators teaching trigonometry, and anyone interested in solving trigonometric equations efficiently.

primarygun
Messages
233
Reaction score
0
General solution for a trigonometry equation.
I solved this equation with several method and I found two possible expressions for the answers. They should be exactly the same. Please help me check for them or combine them together to give the one which is more common. Thanks for any ideas.
\cos x + \sin x=0
x=n\pi -\pi/4
x=(n\pi)/2+\pi/4
 
Last edited:
Mathematics news on Phys.org
Your first is correct.
Letting n be some integer, another way to write the solutions is:
x=\frac{2n+1}{2}\pi+\frac{\pi}{4}
Your last equation is incorrect; set n=2.
This says that x=\frac{5\pi}{4} is a root; but this is untrue, since it lies in the 3.quadrant where both the sine and cosine functions are negative.
 
Oh sorry, I missed to state that the n for the second expression is any odd integer. Really sorry.
 
primarygun said:
Oh sorry, I missed to state that the n for the second expression is any odd integer. Really sorry.
In that case, of course, your second equation is just the one I provided; both 1) and 2) are standard ways of writing the solutions
 
Thank you very much.
 

Similar threads

Undergrad ##1^x =2## has complex solutions?
  • · Replies 7 ·
Replies
7
Views
2K
Graduate Finding Minimal Mean Distance Curves on the Unit Sphere
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Why is it important to convert angle units when using trigonometric functions?
  • · Replies 8 ·
Replies
8
Views
2K
MHB Finding Solutions for $(1+sin^42\theta) = 17(1+sin2\theta)^4$ in $[0,2\pi ]$
  • · Replies 1 ·
Replies
1
Views
987
MHB Solve $2\cos^2(x)=1$: Find $\theta$
  • · Replies 7 ·
Replies
7
Views
2K
High School General explicit solution to polynomial interpolation
  • · Replies 3 ·
Replies
3
Views
2K
MHB Dirac Delta and Fourier Series
  • · Replies 2 ·
Replies
2
Views
2K
MHB Explore the Multiple Grid Method: Get Accurate Solution Faster
  • · Replies 36 ·
2
Replies
36
Views
8K
MHB Linear combination of sine and cosine function
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Why are 0 and pi/2 the solutions for sin(x)=0 and cos(x)=0?
  • · Replies 4 ·
Replies
4
Views
2K