Sketch the graph of $y=f(x)$ for $0.5<x\le 4$, showing the intercepts and the behavior at $x=\frac{1}{2}$.

Solve the inequality $\ln (2x-1)<\sin (x)$.

Show that the equation may be written as $6\cos^2x+\sin x-4=0.$

Express ${\cos }^{2}x$ in terms of $\sin x$, and hence obtain a quadratic equation in $\sin x$.

Solve $6{\sin }^{2}x-\sin x-2=0$ exactly for $\sin x$.

Determine all exact values of $x$ in the interval $0\le x<2\pi$ that satisfy the original equation.

Sketch the graphs of $y=3\cos (2x)$ and $y=1-\sin x$ for $0\le x\le 2\pi$, indicating approximate intersections matching the solutions from part 4.

Show that the equation may be written as $2{\cos }^{2}x-\cos x-1=0$.

Factor the left-hand side to obtain an equation of the form $(2\cos x+1)(\cos x-1)=0$.

Find all exact values of $x$ in the interval $0\le x<2\pi$ that satisfy the equation.

Show that the equation may be written as ${\cos }^{2}x+\frac{3}{2}\sin x=0$.

Express ${\cos }^{2}x$ in terms of $\sin x$, and hence obtain a quadratic equation in $\sin x$.

Solve $2{\sin }^{2}x-3\sin x-2=0$ exactly for $\sin x$.

Determine all exact values of $x$ in the interval $0\le x<2\pi$ that satisfy the original equation.

Show that the equation may be written as $4\cos^2x+1-3\sin x=0.$

Express ${\cos }^{2}x$ in terms of $\sin x$ and obtain a quadratic equation in $\sin x$.

Solve $4{\sin }^{2}x+3\sin x-5=0$ exactly for $\sin x$.

Determine all exact values of $x$ in the interval $0\le x<2\pi$ that satisfy the original equation.