Solving Trigonometric Equations

Basic Equations

Let's take what we just learned and outline some general steps to follow when solving a trigonometric equation.

1. Use algebraic techniques to isolate the \(\sin{x}\), \(\cos{x}\), or \(\tan{x}\) terms. This might include combining like terms, factoring, finding LCD's, etc.
2. Determine the specific solutions on \([0, 2\pi)\) for the equations \(\sin{x} = a\), \(\cos{x} = b\), or \(\tan{x} = c\) using the unit circle. If the \(a\), \(b\), and \(c\) values are not nice unit circle values, then we might need to use technology to compute the corresponding inverse trig value.
3. Express the infinitely many solutions by adding \(2\pi k\) for sine and cosine or \(\pi k\) for tangent.

Note that we will amend this list over the next couple pages as the equations we try to solve become more complicated.

Linear Equations

Let's look at a few examples involving linear terms of \(\sin{x}\), \(\cos{x}\), or \(\tan{x}\).

Non-Linear Equations

Let's look at a few examples involving non-linear terms of \(\sin{x}\), \(\cos{x}\), or \(\tan{x}\), which will probably involve some form of factoring.

Using Technology

Let's look at a few examples involving that include non-unit circle values, which will require the use of technology to evaluate.