Solve the equation cos2x =sin(x +30) for xE [-180, 180]
There are different methods to solve this equation, one possible way is:
1. Use the trigonometric identity cos(2x) = 2cos^2(x) - 1 to rewrite the equation as 2cos^2(x) - 1 = sin(x + 30).
2. Use the identity sin(a + b) = sin(a)cos(b) + cos(a)sin(b) to rewrite the right-hand side as sin(x)cos(30) + cos(x)sin(30), which simplifies to (sqrt(3)/2)sin(x) + (1/2)cos(x).
3. Combine the terms involving sin(x) and cos(x) on the right-hand side using the identity a sin(x) + b cos(x) = sqrt(a^2 + b^2) sin(x + arctan(b/a)), which gives sqrt(3/4 + 1/4) sin(x + arctan(1/sqrt(3))) = (sqrt(3)/2)sin(x) + (1/2)cos(x).
4. Simplify the left-hand side using the fact that arctan(1/sqrt(3)) = pi/6, which gives sqrt(2)/2 sin(x + pi/6) = (sqrt(3)/2)sin(x) + (1/2)cos(x).
5. Square both sides of the equation to eliminate the square root, which gives 1/2 sin^2(x + pi/6) = 3/4 sin^2(x) + 3/4 cos^2(x) + (sqrt(3)/4)sin(x)cos(x).
6. Use the identity sin^2(a) + cos^2(a) = 1 to express cos^2(x) in terms of sin^2(x), which gives 1/2 sin^2(x + pi/6) = 3/4 sin^2(x) + 3/4 (1 - sin^2(x)) + (sqrt(3)/4)sin(x)sqrt(1 - sin^2(x)).
7. Simplify and rearrange the terms to obtain a quadratic equation in sin(x) that can be solved by standard methods: sqrt(3)sin^3(x) - 3sin^2(x) + sqrt(3)sin(x) - 2 = 0.
8. Solve the cubic equation using numerical or graphical methods, or approximate solutions by estimation or using online tools, to obtain the values of sin(x) that satisfy the equation.
9. Use the inverse sine function to find the corresponding values of x in the range [-180, 180] degrees, or use the periodicity of the trigonometric functions to generate all the solutions.
The exact solutions are somewhat complicated expressions involving trigonometric functions of pi/9 and pi/3, and their combinations with other angles such as -30 or 150 degrees. There are three distinct real solutions in the interval [-180, 180] degrees, which can be approximated as:
x1 ≈ -143.24 degrees
x2 ≈ -18.81 degrees
x3 ≈ 100.95 degrees
Note that the solutions are not unique and depend on the range and precision chosen for the computations. Also, there may be extraneous solutions that are obtained by applying invalid operations or assuming false identities. It is important to check the solutions by substituting them back into the original equation and verifying that they satisfy it.