To solve this trigonometric equation for x in the interval [-60, 0], we can use the double angle identities and the Pythagorean identity.
Starting with the given equation: sin^4(x) - cos^4(x) = 1/2
We can rewrite sin^4(x) and cos^4(x) using the double angle identities: (sin^2(x))^2 - (cos^2(x))^2 = 1/2 [(1 - cos^2(x))^2] - cos^4(x) = 1/2 [1 - 2cos^2(x) + cos^4(x)] - cos^4(x) = 1/2 1 - 2cos^2(x) = 1/2 -2cos^2(x) = -1/2 cos^2(x) = 1/4 cos(x) = ±1/2
Since we are looking for solutions in the interval [-60, 0], we can keep only the negative solution: cos(x) = -1/2
The only angle in the given interval where the cosine is equal to -1/2 is -120 degrees or -2π/3. Therefore, the solution to the equation sin^4(x) - cos^4(x) = 1/2 in the interval [-60, 0] is x = -2π/3.
To solve this trigonometric equation for x in the interval [-60, 0], we can use the double angle identities and the Pythagorean identity.
Starting with the given equation:
sin^4(x) - cos^4(x) = 1/2
We can rewrite sin^4(x) and cos^4(x) using the double angle identities:
(sin^2(x))^2 - (cos^2(x))^2 = 1/2
[(1 - cos^2(x))^2] - cos^4(x) = 1/2
[1 - 2cos^2(x) + cos^4(x)] - cos^4(x) = 1/2
1 - 2cos^2(x) = 1/2
-2cos^2(x) = -1/2
cos^2(x) = 1/4
cos(x) = ±1/2
Since we are looking for solutions in the interval [-60, 0], we can keep only the negative solution:
cos(x) = -1/2
The only angle in the given interval where the cosine is equal to -1/2 is -120 degrees or -2π/3. Therefore, the solution to the equation sin^4(x) - cos^4(x) = 1/2 in the interval [-60, 0] is x = -2π/3.