Introduction to Trigonometric Equations

Welcome to one of the most rewarding parts of trigonometry! So far, you have learned about the unit circle, graphs, and identities. Now, we are going to put all those pieces together to solve Trigonometric Equations. Solving a trigonometric equation is simply finding the value of an angle (usually \(x\) or \(\theta\)) that makes the equation true. For example, if we have \(\sin x = 0.5\), we are looking for the angles where the "height" on the unit circle is exactly \(0.5\). Don't worry if this seems tricky at first. Unlike regular algebra where \(x + 5 = 10\) has only one answer, trigonometric equations often have multiple answers because the functions repeat themselves. We will learn how to find all the correct "hidden" answers within a specific range.

1. Understanding the Finite Interval

In the IB syllabus, you are required to solve equations within a finite interval (a specific range). You won't be asked for "general solutions" that go on forever. Common intervals include:
  • \(0 \le x \le 2\pi\) (Radians)
  • \(0^\circ \le x \le 360^\circ\) (Degrees)
  • \(-\pi \le x \le \pi\)
Quick Tip: Always check your calculator mode first! If the interval has a \(\pi\) in it, you must be in Radian mode. If it has a \(^\circ\) symbol, you must be in Degree mode.

2. Solving Simple Linear Equations

A linear trigonometric equation looks like \(\sin x = k\), \(\cos x = k\), or \(\tan x = k\).

The Step-by-Step Process:

  1. Isolate the trig function: Get the \(\sin\), \(\cos\), or \(\tan\) by itself on one side.
  2. Find the Principal Value: Use your calculator (\(\sin^{-1}\)) or your knowledge of exact values (like \(\frac{\pi}{6}\) or \(45^\circ\)) to find the first angle.
  3. Find the Secondary Values: Use the unit circle or the CAST diagram to find other angles in the interval that have the same value.
Example:
Solve \(2\cos x - \sqrt{3} = 0\) for \(0 \le x \le 2\pi\).
Step 1: Isolate \(\cos x\).

\(2\cos x = \sqrt{3} \implies \cos x = \frac{\sqrt{3}}{2}\)

Step 2: Find the first angle. We know from exact values that \(\cos \frac{\pi}{6} = \frac{\sqrt{3}}{2}\). So, \(x_1 = \frac{\pi}{6}\).
Step 3: Find the second angle. Cosine is positive in Quadrant 1 and Quadrant 4.

In Quadrant 4: \(x_2 = 2\pi - \frac{\pi}{6} = \frac{11\pi}{6}\).

Final Answer: \(x = \frac{\pi}{6}, \frac{11\pi}{6}\).

3. Solving Equations Leading to Quadratics

Sometimes, a trigonometric equation looks like a quadratic equation (\(ax^2 + bx + c = 0\)). These usually involve \(\sin^2 x\), \(\cos^2 x\), or \(\tan^2 x\).

How to spot them:

Look for the squared symbol. For example: \(2\sin^2 x + \sin x - 1 = 0\).

The "Substitution Trick":

To make it easier, replace the trig function with a letter like \(u\).

Let \(u = \sin x\). The equation becomes: \(2u^2 + u - 1 = 0\).

Now, factorize it like a normal quadratic:

\((2u - 1)(u + 1) = 0\)

This gives us two smaller equations to solve:
  1. \(u = \frac{1}{2} \implies \sin x = \frac{1}{2}\)
  2. \(u = -1 \implies \sin x = -1\)
Then, find all possible values for \(x\) in the given interval for both equations!

4. Solving Graphically (Paper 2 & 3)

On calculator papers, you can solve complex equations by graphing. This is often the fastest and most reliable method.

Using your GDC (Graphic Display Calculator):

  1. Set the left side of the equation as \(y_1\).
  2. Set the right side of the equation as \(y_2\).
  3. Adjust your Window settings to match the interval given in the question (e.g., \(x_{min} = 0\), \(x_{max} = 2\pi\)).
  4. Use the Intersect tool to find the \(x\)-coordinates of every point where the lines cross.
Common Mistake: Forgetting to change the \(y\)-axis in your window. If the equation is \(\sin x = 5\), but your screen only goes from \(-1\) to \(1\), you won't see the intersection! (Though in that case, there would be no solution).

5. Important Identities to Remember

Sometimes you need to "clean up" an equation before you can solve it. These identities (from previous chapters) are your best friends:
  • The Tangent Identity: \(\tan \theta = \frac{\sin \theta}{\cos \theta}\) (Useful if you have both \(\sin\) and \(\cos\) in an equation).
  • The Pythagorean Identity: \(\cos^2 \theta + \sin^2 \theta = 1\). If an equation has both \(\sin^2 x\) and \(\cos x\), use this to get everything in terms of \(\cos\).
  • Double Angle Formulae: \(\sin 2\theta = 2\sin \theta \cos \theta\) and \(\cos 2\theta = \cos^2 \theta - \sin^2 \theta\) (or its other forms).

Summary & Key Takeaways

  • Intervals Matter: Always check if the answer needs to be in degrees or radians and stay within the bounds.
  • CAST Diagram / Unit Circle: Most trig equations have at least two solutions in one full circle (\(2\pi\) or \(360^\circ\)). Don't stop at the first answer your calculator gives you!
  • Quadratic Style: If you see a squared trig function, try to factorize it like a quadratic equation.
  • GDC is a Power Tool: For Paper 2, use the "Intersect" function to save time and avoid algebraic errors.

Quick Review: Common Pitfalls
1. Wrong Mode: Solving a radian problem in degree mode is the most common way to lose easy marks.
2. Missing Solutions: If the equation is \(\sin(2x) = 0.5\), the "frequency" has doubled. This usually means there are twice as many solutions in the same interval. Always check!
3. Dividing by a Function: Never divide both sides by \(\sin x\) or \(\cos x\), as you might "cancel out" and lose valid solutions. Instead, move everything to one side and factorise.