91Ó°ÊÓ

We are given the equation: $$5 \sin 3x + 6 \cos 3x = 1$$ We can rewrite this equation in the form R sin(3x + α) using the angle addition formula for sine. Let R sin(α) = 5, and R cos(α) = 6. Then, the equation becomes: $$R(\sin 3x \cos \alpha + \cos 3x \sin \alpha) = 1$$ Solve for R and α: $$R = \sqrt{(5)^2 + (6)^2} = \sqrt{61}$$ $$\tan \alpha = \frac{5}{6} \Rightarrow \alpha = \arctan\left(\frac{5}{6}\right)$$ Now, the equation becomes: $$\sqrt{61} \sin(3x + \arctan(\frac{5}{6})) = 1$$

Now, we will simplify the equation by dividing both sides by the R value: $$\sin(3x + \arctan(\frac{5}{6})) = \frac{1}{\sqrt{61}}$$

Now that we have expressed the equation in terms of a single trigonometric function, we can solve the equation graphically by plotting the function and finding the points where it intersects the line at the given value. To find the solution graphically, we can plot the left-hand side of the simplified equation: $$ f(x) = \sin(3x + \arctan(\frac{5}{6}))$$ And the right-hand side constant: $$g(x) = \frac{1}{\sqrt{61}}$$ The points where f(x) intersects g(x) graphically will provide the solutions for x. Remember that sine functions have a periodic nature, so you might find multiple intersections within a specified interval. If you want to find all solutions within a given range, graph the functions within that range and look for their intersections points.