Welcome to Trigonometric Equations!

Ever wondered how your smartphone processes sound waves, or how engineers predict the rise and fall of the tides? They use trigonometric equations. In this chapter, we aren't just looking at triangles anymore; we are looking at the beautiful, repeating patterns of the sine, cosine, and tangent functions to solve for unknown angles.

If you have ever solved a basic equation like \(2x - 4 = 0\), you already have the foundation. We are just replacing \(x\) with things like \(\sin \theta\). Don’t worry if it feels a bit "wavy" at first—we will break it down step-by-step!

1. The Toolkit: What You Need to Know First

Before we dive into solving, you need two main tools in your belt: Exact Values and Identities.

A. Essential Exact Values

You are expected to know the exact values for \(0^\circ, 30^\circ, 45^\circ, 60^\circ,\) and \(90^\circ\). While you can use a calculator, knowing these helps you spot patterns in harder questions!

  • \(\sin 30^\circ = 0.5\)
  • \(\cos 60^\circ = 0.5\)
  • \(\tan 45^\circ = 1\)

B. The Golden Identities

To solve complex equations, we often need to "simplify" them into a single trig ratio using these two identities:

  1. The Tangent Identity: \(\tan \theta \equiv \frac{\sin \theta}{\cos \theta}\)
  2. The Pythagorean Identity: \(\sin^2 \theta + \cos^2 \theta \equiv 1\)

Quick Review: Think of identities as "translation rules." If an equation has both \(\sin^2 \theta\) and \(\cos \theta\), you can use the second identity to turn everything into \(\cos \theta\)!

2. Solving Simple Linear Trig Equations

A linear trig equation looks like this: \(\sin \theta = 0.5\). Your goal is to find all the angles within a specific range (usually \(0^\circ \le \theta < 360^\circ\)) that make this true.

The Step-by-Step Process:

  1. Find the Principal Value (PV): Use your calculator. For \(\sin \theta = 0.5\), type \(\sin^{-1}(0.5)\). The calculator gives you \(30^\circ\).
  2. Find the Secondary Values: Trig functions are periodic (they repeat!). Because of the symmetry of the graphs, there is usually another answer.
  3. Check the Range: Make sure your answers are within the interval requested (e.g., \(0\) to \(360\)).

How to find that "Second Answer": The CAST Diagram

The CAST diagram helps you remember where each ratio is positive:

  • Quadrant 1 (0-90°): All are positive.
  • Quadrant 2 (90-180°): Sine is positive. (\(180 - PV\))
  • Quadrant 3 (180-270°): Tangent is positive. (\(180 + PV\))
  • Quadrant 4 (270-360°): Cosine is positive. (\(360 - PV\))

Memory Aid: C-A-S-T can be remembered as "All Students Take Coffee" (starting from the top right and going counter-clockwise).

Example: Solve \(\sin \theta = 0.5\) for \(0 \le \theta < 360^\circ\).
\(PV = 30^\circ\). Since sine is positive, the second answer is in the S quadrant: \(180 - 30 = 150^\circ\).
Solutions: \(\theta = 30^\circ, 150^\circ\).

3. Working with Multiple Angles

Sometimes you’ll see equations like \(\tan 3\theta = -1\). This means the wave is "squashed"—it repeats 3 times as fast!

The Trick: Expand the Range!
If the question asks for solutions where \(-180^\circ < \theta < 180^\circ\), but the angle is \(3\theta\), you must look for values of \(3\theta\) between \(-540^\circ\) and \(540^\circ\).

Steps for Multiple Angles:
1. Let \(X = 3\theta\). Solve \(\tan X = -1\) as usual.
2. Find all possible values for \(X\) in the expanded range by adding/subtracting \(180^\circ\) (for tan) or \(360^\circ\) (for sin/cos).
3. Finally, divide all your answers by 3 to find \(\theta\).

Common Mistake: Don't divide by 3 too early! Find all your "fake" angles (\(X\)) first, then divide at the very end.

4. Quadratic Trig Equations

If you see a \(\sin^2 \theta\), you are dealing with a quadratic. These usually look like: \(6\sin^2 \theta + \cos \theta - 4 = 0\).

Step 1: Get it into one "language."
We can't solve an equation with both \(\sin\) and \(\cos\) easily. Use \(\sin^2 \theta = 1 - \cos^2 \theta\) to swap the \(\sin^2 \theta\) out.

Step 2: Substitute.
Let \(y = \cos \theta\). Now you have a normal quadratic: \(6(1 - y^2) + y - 4 = 0\).
Rearrange: \(6y^2 - y - 2 = 0\).

Step 3: Factorise and Solve.
Solve for \(y\) (you might get two values, like \(y = 2/3\) and \(y = -1/2\)).

Step 4: Solve for \(\theta\).
Now solve \(\cos \theta = 2/3\) and \(\cos \theta = -1/2\) using the CAST diagram methods we learned earlier.

Did you know? Sometimes a quadratic gives you a value like \(\sin \theta = 2\). Since the sine wave never goes above 1, this part has no solutions. Just write "no solutions" for that branch and move on!

5. Summary and Key Takeaways

Key Terms:

  • Principal Value: The first answer your calculator gives you.
  • Periodicity: The fact that trig graphs repeat every \(360^\circ\) (or \(180^\circ\) for tan).
  • Identity: An equation that is always true, used for substitution.

Top Tips for Success:

  • Always check your range at the end.
  • Draw the graph or CAST diagram for every question—don't try to do it all in your head!
  • Don't divide by a trig ratio. If you have \(\sin \theta \cos \theta = \sin \theta\), don't divide by \(\sin \theta\) (you'll lose solutions!). Instead, factorise: \(\sin \theta (\cos \theta - 1) = 0\).

Don’t worry if this seems tricky at first! Trigonometry is all about practice and spotting which "tool" to use from your toolkit. Keep going!