Introduction to Trigonometric Equations and Inequalities

Welcome! In the previous chapters, you’ve learned how trigonometric functions behave and how their inverse functions work. Now, it’s time to put those skills to work. Solving a trigonometric equation is like being a detective: you are trying to find the specific angle \(x\) (usually in radians!) that makes a statement true. Because trigonometric functions are periodic (they repeat their values forever), these equations can sometimes have an infinite number of solutions. Don't worry if this seems a bit overwhelming at first—we will break it down step-by-step!

Why does this matter? Trigonometric equations help us predict when a tide will be high enough for a boat to dock, or at what exact moments a pendulum will reach its highest point. If it repeats, trig equations can solve it!

1. Solving Basic Trigonometric Equations

The simplest trig equations look like \( \sin(x) = c \), \( \cos(x) = c \), or \( \tan(x) = c \). To solve these without a calculator, you need to be comfortable with your Unit Circle values.

Finding Solutions on a Restricted Interval

Often, a problem will ask for solutions only within one full rotation, such as \( [0, 2\pi) \).

Example: Solve \( 2\cos(x) + 1 = 0 \) for \( 0 \leq x < 2\pi \).

  1. Isolate the trig function: \( 2\cos(x) = -1 \implies \cos(x) = -\frac{1}{2} \).
  2. Identify the quadrants: Since cosine is negative, our angle \(x\) must be in Quadrant II or Quadrant III.
  3. Find the angles: On the unit circle, \( \cos(x) = -\frac{1}{2} \) at \( x = \frac{2\pi}{3} \) and \( x = \frac{4\pi}{3} \).

The General Solution

If the problem does not give you an interval, you must account for the fact that the function repeats every period. We do this by adding \( 2\pi n \) (for sine and cosine) or \( \pi n \) (for tangent), where \( n \) is any integer.

Example: Find all solutions for \( \sin(x) = \frac{\sqrt{2}}{2} \).

\( x = \frac{\pi}{4} + 2\pi n \) and \( x = \frac{3\pi}{4} + 2\pi n \).

Quick Tip: Always check if the question asks for solutions in a specific interval or "all real solutions." That "plus \( 2\pi n \)" part is the most common thing students forget on exams!

2. Algebraic Strategies for Complex Equations

Sometimes, equations look more like the algebra problems you solved in Unit 1. You can use the same techniques: factoring and substitution.

Factoring to Solve

If you see a "squared" trig term, try to set the equation to zero and factor it like a quadratic.

Example: Solve \( 2\sin^2(x) - \sin(x) = 0 \).

  1. Factor out the common term: \( \sin(x)(2\sin(x) - 1) = 0 \).
  2. Set each factor to zero: \( \sin(x) = 0 \) or \( 2\sin(x) - 1 = 0 \).
  3. Solve for \(x\):
    For \( \sin(x) = 0 \), \( x = 0, \pi, 2\pi, \dots \)
    For \( \sin(x) = \frac{1}{2} \), \( x = \frac{\pi}{6}, \frac{5\pi}{6}, \dots \)

Using Identities

As covered in Topic 3.12, you may need to use Pythagorean Identities (like \( \sin^2(x) + \cos^2(x) = 1 \)) to make the entire equation use the same trig function before you start solving. However, remember that for Topic 3.10, the focus is on the solving process itself.

Key Takeaway: Treat \( \sin(x) \) or \( \cos(x) \) like a single variable (like \(u\)). If the equation looks like \( 2u^2 - u = 0 \), solve for \(u\) first, then find the angle \(x\).

3. Solving with Technology (Graphing Calculator)

On the AP Exam, Section I Part B and Section II Part A allow the use of a graphing calculator. Some equations are "analytically inaccessible" (meaning they are too hard to solve by hand).

How to solve using your calculator:

  1. The Intersection Method: Set \( y_1 \) equal to the left side of the equation and \( y_2 \) equal to the right side. Use the "Intersect" feature to find where the graphs cross.
  2. The Zero Method: Move everything to one side so the equation equals zero. Set \( y_1 \) to that expression and use the "Zero" or "Root" feature to find where the graph touches the x-axis.

Important Convention: Per the AP Precalculus CED, you must keep your calculator in radian mode. Also, when reporting decimal answers, always round or truncate to at least three decimal places (e.g., \( x \approx 0.844 \)).

4. Trigonometric Inequalities

Trigonometric inequalities ask when a trig function is "greater than" or "less than" a certain value. These are best understood by looking at a graph.

Example: Solve \( \sin(x) > \frac{1}{2} \) on the interval \( [0, 2\pi] \).

  1. Find the boundaries: We know \( \sin(x) = \frac{1}{2} \) at \( x = \frac{\pi}{6} \) and \( x = \frac{5\pi}{6} \).
  2. Visualize the graph: Look at the sine wave. Between \( \frac{\pi}{6} \) and \( \frac{5\pi}{6} \), the "hump" of the graph is above the line \( y = 0.5 \).
  3. Write the interval: The solution is \( (\frac{\pi}{6}, \frac{5\pi}{6}) \).

Did you know? You can use "test points" just like you did with polynomial inequalities. Pick a value in an interval, plug it into the inequality, and see if it's true!

5. Common Mistakes to Avoid

  • Dividing by a variable: Never divide both sides of an equation by \( \sin(x) \) or \( \cos(x) \). You might "cancel out" and lose a valid solution! Always factor instead.
  • Ignoring the Domain: Pay close attention to whether the answer should be in degrees or radians (AP Precalculus defaults to radians) and what the specified interval is.
  • Calculator Errors: Make sure you are in Radian mode. If your graph looks like a flat line, check your "Window" settings!

Quick Review: The Strategy Checklist

  1. Is it a "No Calculator" or "Calculator" question?
  2. Can I isolate the trig function or do I need to factor?
  3. Do I need a general solution (\( + 2\pi n \)) or just values on an interval?
  4. If it's an inequality, have I identified the "boundary" points first?
  5. Are my decimal answers accurate to three places?

Keep practicing! Solving these equations is a fundamental skill that connects everything you’ve learned about circles, functions, and algebra. You've got this!