To solve the equation cos(x + π/6) + sin(x - π/3) = 0, you can use basic algebraic operations to simplify the equation and solve for the variable x.
Let's break it down step by step:
Step 1: Simplify the equation using trigonometric identities.
Using the sum and difference formulas for cosine and sine, we can rewrite the equation as:
cos(x)cos(Ï€/6) - sin(x)sin(Ï€/6) + sin(x)cos(Ï€/3) - cos(x)sin(Ï€/3) = 0
Simplifying further:
(√3/2)cos(x) - (1/2)sin(x) + (1/2)sin(x) - (√3/2)cos(x) = 0
Step 2: Combine like terms.
The cos(x) terms and sin(x) terms cancel each other out:
0 = 0
Step 3: Interpret the result.
The equation simplifies to 0 = 0, which means that the equation is satisfied for all values of x. Therefore, there are infinitely many solutions to this equation.
In conclusion, for the given equation cos(x + π/6) + sin(x - π/3) = 0, all values of x will satisfy the equation, since 0 is always equal to 0.