Final answer:
To find the smallest positive solution to the equation sin(x) = cos(2x+π/3), we can rewrite it as sin(x) - cos(2x+π/3) = 0. By applying trigonometric identities, simplifying and solving for x, the smallest positive solution is x ≈ 330° (or 11π/6 radians).
Step-by-step explanation:
To find the smallest positive solution to the equation sin(x) = cos(2x+π/3), we can rewrite it as sin(x) - cos(2x+π/3) = 0. Then, using a trigonometric identity, we can express cos(2x+π/3) in terms of sin(x): cos(2x+π/3) = -sin(x+π/6). Substituting this back into the equation gives sin(x) + sin(x+π/6) = 0.
Now, we can use the sum-to-product formula for sine to simplify further: sin(x) + sin(x)cos(π/6) + cos(x)sin(π/6) = 0.
Rearranging the terms gives sin(x) + (√3/2)sin(x) + (1/2)cos(x) = 0. Using the identity sin(x) = cos(π/2 - x), we can simplify this to (√3/2)cos(π/2 - x) + (1/2)cos(x) = 0.
Next, let's convert degrees to radians by substituting π/6 with 30°.
Now the equation becomes (√3/2)cos(pi/2 - x) + (1/2)cos(x) = 0.
Expanding the equation gives (√3/2)sin(x) + (1/2)cos(x) = 0.
Multiply every term by 2 to eliminate the fractions: √3sin(x) + cos(x) = 0.
Rearranging the terms gives √3sin(x) = -cos(x).
Finally, we can solve for x by taking the inverse sine of both sides: x = sin⁻¹(-1/√3).
Evaluating this gives x ≈ -30° (or -π/6 radians).
However, since we need to find the smallest positive solution, we can add 360° (or 2π radians) to get the answer in the desired range: x ≈ 330° (or 11π/6 radians).