Trigonometric equations

Welcome to Trigonometric Equations!

Hi there! In this chapter, we are going to learn how to find the "unknown" angles in equations involving sine, cosine, and tangent. Solving these is a bit like solving a puzzle: we want to find all the possible values of \(\theta\) (theta) that make an equation true.

Why does this matter? Trigonometry isn't just about triangles; it’s the mathematics of anything that repeats in cycles—like sound waves, the movement of the tides, or even the vibration of a guitar string. Let’s get started!

1. The Basics: Solving Simple Equations

A simple trigonometric equation looks something like this: \(\sin \theta = 0.5\). To solve it, we need to find the angle \(\theta\). However, because trigonometric graphs repeat forever, there are usually multiple answers within a specific interval (like \(0^\circ \leq \theta < 360^\circ\)).

Step-by-Step: Finding Your Solutions

1. The Principal Value (PV): Use your calculator to find the first angle. For \(\sin \theta = 0.5\), you do \(\sin^{-1}(0.5)\), which gives \(30^\circ\).
2. The Interval: Check if the question asks for Degrees or Radians (\(\pi\)). Make sure your calculator is in the correct mode!
3. Finding Other Values: Trigonometric functions are symmetrical. You can use the unit circle or the function graphs to find other solutions.

Memory Aid: The CAST Diagram

To remember which functions are positive in which quadrant, use the CAST diagram (starting from the bottom-right and going counter-clockwise):
- Cosine is positive in the 4th quadrant (\(270^\circ\) to \(360^\circ\)).
- All are positive in the 1st quadrant (\(0^\circ\) to \(90^\circ\)).
- Sine is positive in the 2nd quadrant (\(90^\circ\) to \(180^\circ\)).
- Tangent is positive in the 3rd quadrant (\(180^\circ\) to \(270^\circ\)).

Mnemonic: Castles Are So Tall or Add Sugar To Coffee.

Quick Review: To find the second value for \(\sin \theta = k\), use \(180^\circ - \text{PV}\). For \(\cos \theta = k\), use \(360^\circ - \text{PV}\). For \(\tan \theta = k\), use \(180^\circ + \text{PV}\).

2. Working with Multiple Angles

Sometimes you’ll see equations like \(\tan 3\theta = -1\). Don't worry if this looks tricky! Think of \(3\theta\) as a single block for a moment.

The "Adjust the Range" Trick

If the interval is \(0^\circ \leq \theta < 180^\circ\), but the angle is \(3\theta\), you must multiply your range by 3. So, you look for all solutions between \(0^\circ\) and \(540^\circ\).
1. Solve for the "block": \(3\theta = \tan^{-1}(-1)\).
2. Find all values for \(3\theta\) in the new wider range.
3. Divide all your final answers by 3 to get \(\theta\).

Common Mistake: Students often find one value for \(\theta\) and then multiply it. Always find all the values for the multiple angle first, and then divide at the very end!

3. Using Identities to Simplify

Often, an equation has a mix of different trig functions. We use identities to turn everything into the same type (e.g., all sines or all cosines).

The Big Two Identities

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

Example: Solve \(6\sin^2 \theta + \cos \theta - 4 = 0\).
Since we have a \(\cos \theta\), let’s change \(\sin^2 \theta\) into \(1 - \cos^2 \theta\).
\(6(1 - \cos^2 \theta) + \cos \theta - 4 = 0\)
\(6 - 6\cos^2 \theta + \cos \theta - 4 = 0\)
\(6\cos^2 \theta - \cos \theta - 2 = 0\)

This is now a Quadratic Equation! You can treat \(\cos \theta\) like \(x\) and solve using factorisation or the quadratic formula.

4. Advanced Identities (Stage 2)

For higher-level problems, you will use the reciprocal identities and double-angle formulae.

Reciprocal Identities

- \(\sec^2 \theta \equiv 1 + \tan^2 \theta\)
- \(\text{cosec}^2 \theta \equiv 1 + \cot^2 \theta\)

Did you know? These are just variations of \(\sin^2 \theta + \cos^2 \theta = 1\). If you divide the whole thing by \(\cos^2 \theta\), you get the \(\sec\) identity!

Double Angle Formulae

These are great for equations where one part is \(\sin 2\theta\) and the other is \(\sin \theta\).
- \(\sin 2\theta \equiv 2\sin \theta \cos \theta\)
- \(\cos 2\theta \equiv \cos^2 \theta - \sin^2 \theta\) (which can also be written as \(2\cos^2 \theta - 1\) or \(1 - 2\sin^2 \theta\)).

Key Takeaway: If you see a "2" inside the function (like \(\cos 2\theta\)), look to use a double-angle formula to break it down into single \(\theta\) terms.

5. The Harmonic Form: \(R \cos(\theta \pm \alpha)\)

If you see an equation like \(3\cos \theta + 4\sin \theta = 2\), you can't easily use sines or cosines alone. Instead, we squash them together into a single wave: \(R \cos(\theta - \alpha)\) or \(R \sin(\theta + \alpha)\).

1. Find \(R\) using Pythagoras: \(R = \sqrt{a^2 + b^2}\).
2. Find \(\alpha\) using \(\tan \alpha = \frac{b}{a}\).
3. Rewrite the equation: \(R \cos(\theta - \alpha) = c\).
4. Solve for \((\theta - \alpha)\) just like we did with multiple angles!

6. Summary of Common Pitfalls

Don't get caught out by these!

• Dividing by a trig function: Never divide an equation by \(\sin \theta\) or \(\cos \theta\) to cancel it out. You might be "dividing by zero" and losing a whole set of solutions! Instead, factorise it out.
• The \(\pm\) sign: When taking a square root (e.g., \(\cos^2 \theta = 0.25\)), remember that \(\cos \theta\) could be \(+0.5\) or \(-0.5\).
• Radians vs Degrees: Always check your calculator at the start of every question. \(\pi\) means Radians!

Key Takeaway: Solving trigonometric equations is a process of: Simplify (using identities), Solve (to find the Principal Value), and Expand (finding all other angles in the range using symmetry).

Don't worry if this seems tricky at first. Trigonometry is very visual—try sketching the graphs of \(\sin\), \(\cos\), and \(\tan\) whenever you feel stuck!