how would you solve: sin(2x)+cos(2x)=tan(x)?
sin(2x)+cos(2x)=tan(x)
cos(2x) = sinx/cosx - 2sinxcosx
cos(2x) = (sinx - 2sinxcos^2x)/cosx
cos(2x) = sinx(1 - 2cos^2x)/cosx
cos(2x) = tanx(-cos(2x))
-1 = tanx
x = 135º or x = 315º
or in radians
x = 3pi/4 or 7pi/4
2 sinx cosx + cos^2x - sin^2x
= sinx/cosx
2 sinx cos^2x + cos^2x - sin^2x = sin x
2 sinx(1-sin^2x)+ (1-2sin^2x) = sinx
Treat sinx as the variable (u) and solve the polynomial
2 u(1 - u^2)+ 1 - 2u^2 = u
-2u^3 -2u^2 + u +1 = 0
My answer is wrong in the second line and what follows. Go with Reiny's
To solve the equation sin(2x) + cos(2x) = tan(x), we need to use trigonometric identities and algebraic manipulations. Here's how we can approach it:
Step 1: Simplify the equation using trigonometric identities.
The double-angle formula for sine is sin(2x) = 2sin(x)cos(x).
The double-angle formula for cosine is cos(2x) = cos^2(x) - sin^2(x).
The formula for tangent is tan(x) = sin(x)/cos(x).
Substituting these identities into the equation, we have:
2sin(x)cos(x) + cos^2(x) - sin^2(x) = sin(x)/cos(x).
Step 2: Simplify the equation further.
Rearrange the equation:
2sin(x)cos(x) + cos^2(x) - sin^2(x) - sin(x)/cos(x) = 0.
Multiply both sides of the equation by cos(x):
2sin(x)cos^2(x) + cos^3(x) - sin^2(x)cos(x) - sin(x) = 0.
Step 3: Combine similar terms.
Rearrange the equation:
2sin(x)cos^2(x) - sin^2(x)cos(x) + cos^3(x) - sin(x) = 0.
Step 4: Factor out sin(x).
sin(x)(2cos^2(x) - sin(x)cos(x) + cos^2(x) - 1) = 0.
The first solution is sin(x) = 0, which gives us x = nπ, where n is an integer.
Now, let's solve the other factor:
2cos^2(x) - sin(x)cos(x) + cos^2(x) - 1 = 0.
Step 5: Rearrange the equation.
3cos^2(x) - sin(x)cos(x) - 1 = 0.
Step 6: Solve the quadratic equation.
Let's make a substitution. Let t = cos(x).
3t^2 - t√(1 - t^2) - 1 = 0.
This is a quadratic equation in t, which can be solved using various methods such as factoring, completing the square, or the quadratic formula.
Once you find the values of t, substitute them back into the equation t = cos(x) and solve for x.
Remember to check for extraneous solutions, as some values of x may not satisfy the original equation sin(2x) + cos(2x) = tan(x).
Note: The solution to this equation can involve a complex calculation process, so it's recommended to use numerical methods or graphing calculators to get accurate results.