Solve the triginometric equation analytically. Use values of x for 0 less then or equal to x less then 2pi
The equation for this question is sin2x + cos2x = 0
how about
sin2x = -cos2x, now divide by cos2x
sin2x/cos2x = -1
tan2x = -1
so 2x = 135 or 315º
so x = 67.5 or x = 157.5º
now remember that the period of tan(kß) is 180/k
so the period of tan2x is 90º, which means that if we add 90º to any of our answers we get a new answer as long as we are in our given domain.
so x = 67.5, 157.5, 247.5, 337.5
again, it would be a good idea to check these with your calculator.
To solve the trigonometric equation analytically, we can start by using the trigonometric identity:
sin^2(x) + cos^2(x) = 1
We have a similar equation, but with a negative sign instead of 1:
sin^2(x) + cos^2(x) = 0
This means that sin^2(x) and cos^2(x) both have to equal 0. For sin^2(x) = 0, the only solution is sin(x) = 0. Similarly, for cos^2(x) = 0, the only solution is cos(x) = 0.
Using the unit circle, we can easily determine the angles for which sin(x) = 0 and cos(x) = 0 within the given domain of 0 ≤ x < 2π.
For sin(x) = 0, the solutions are x = 0, π, and 2π. These angles correspond to the points where the unit circle intersects the x-axis.
For cos(x) = 0, the solutions are x = π/2 and 3π/2. These angles correspond to the points where the unit circle intersects the y-axis.
Therefore, the solutions to the equation sin^2(x) + cos^2(x) = 0 within the given domain are x = 0, π/2, π, 3π/2, and 2π.