Solve : 2tan3x+cos2x+1=tan3x+2cos2x
kind of messy. First, just rearrange things a bit to get
tan3x + 1 = cos2x
this is clearly true for multiples of 2π, but there are surely other roots
I doubt this will yield to algebraic solution. The resulting attempts at simplification will result in cubic equations, so you better trot out your graphic or numeric methods.
I get other roots as x = 0.78, 1.73, 3.93, 4.87
To solve the equation 2tan(3x) + cos(2x) + 1 = tan(3x) + 2cos(2x), you can follow these steps:
Step 1: Combine like terms
- Start by bringing all the terms involving tan(3x) to one side and the terms involving cos(2x) to the other side.
- Subtract tan(3x) from both sides: 2tan(3x) + cos(2x) + 1 - tan(3x) = 2cos(2x)
- Simplify the equation: tan(3x) + cos(2x) + 1 = 2cos(2x)
Step 2: Express cos(2x) in terms of tan(3x)
- Use the identity cos(2x) = 1 - tan^2(x) to express cos(2x) in terms of tan(x).
- Substitute this expression into the equation: tan(3x) + (1 - tan^2(x)) + 1 = 2(1 - tan^2(x))
Step 3: Simplify and solve for tan(x)
- Expand the expression: tan(3x) + 1 - tan^2(x) + 1 = 2 - 2tan^2(x)
- Rearrange the terms: -2tan^2(x) + tan(3x) - tan^2(x) = 0
- Combine the terms: -3tan^2(x) + tan(3x) = -2
Step 4: Apply the trigonometric identity
- Use the identity tan(3x) = (3tan(x) - tan^3(x))/(1 - 3tan^2(x)) to express tan(3x) in terms of tan(x).
- Substitute the expression into the equation: -3tan^2(x) + (3tan(x) - tan^3(x))/(1 - 3tan^2(x)) = -2
Step 5: Solve the resulting quadratic equation
- Multiply both sides by (1 - 3tan^2(x)) to clear fractions: -3tan^2(x)(1 - 3tan^2(x)) + (3tan(x) - tan^3(x)) = -2(1 - 3tan^2(x))
- Expand and rearrange the terms: 3tan^4(x) - 9tan^2(x) + 3tan(x) - tan^3(x) - 2 + 6tan^2(x) = 0
- Collect like terms: 3tan^4(x) - tan^3(x) - 3tan^2(x) + 3tan(x) - 2 = 0
At this point, the equation becomes a quartic equation in terms of tan(x). You can attempt to factor, use numerical methods, or approximate solutions graphically to find the values of tan(x) that satisfy the equation.