Exams · Russian state exam, profile level

No. 13 An equation with roots on a segment

A trigonometric equation from part 2: solve it and pick the roots that lie on a segment.

How to solve it

Trigonometric equations with roots on a segment

This is a part 2 task: solve a trigonometric equation such as $2\cos^2 x - \cos x - 1 = 0$, then pick the roots that lie on a given segment. The equation has infinitely many solutions, the segment holds only a few.

Step by step

Reduce the equation to one function. For example, replace $\sin^2 x$ with $1 - \cos^2 x$ so that only cosine remains.
Substitute $t = \cos x$ and solve an ordinary, often quadratic, equation. Remember that cosine and sine lie between $-1$ and $1$: drop any other values of $t$.
Go back to $x$ with the formulas for the basic equations.
Pick the roots on the segment: try integer $n$ one by one, or mark the points on the unit circle.

A number from $-1$ to $1$. Outside this range there are no roots.

Any integer: the roots repeat every full turn.

Example: $\cos x = \frac{1}{2}$ on $[0, 2\pi]$. The general solution is $x = \pm\frac{\pi}{3} + 2\pi n$; the segment contains $\frac{\pi}{3}$ and $\frac{5\pi}{3}$.

Common mistakes

Keeping a value of $t$ outside $[-1, 1]$ and hunting for roots that do not exist.
Forgetting the second series $-\arccos a + 2\pi n$ and losing half the roots.
Missing roots at the ends of the segment. The ends belong to it.