Solving Trigonometric Equations

Multi-Angle Equations

The last piece that we need to consider is solving equations where the inside of the trig function is not just \(x\). For example, how would we solve an equation containing an expression like \(\sin(3\theta)\) or \(\cos(\pi \theta + 1)\)? We can again follow the previous outline with an extra modification. The main difference here will be that we can replace the inner expression with an \(x\), solve the trig equation for \(x\) like we have already been doing, and then finish by solving the \(x\) equation for \(\theta\).

1. Use trigonometric identities and 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 for \(x\) by adding \(2\pi k\) for sine and cosine or \(\pi k\) for tangent.
4. If this is a multi-angle problem, then replace \(x\) with the corresponding expression containing \(\theta\) and use algebraic techniques to solve for \(\theta\). List the infinitely many solutions for \(\theta\) and any specific solutions on \([0, 2\pi)\).

This may sound rather complicated, but it is not much different that what we have already been doing. Also, realize that the variables used in this outline may not directly match the variables in a given equation. Let's look at a few examples.