Welcome to Trigonometric Identities!
Hello! Welcome to one of the most powerful chapters in A Level Mathematics. If you’ve ever looked at a long, messy equation and wished you could just wave a magic wand to make it simpler, you’re in the right place. Trigonometric identities are exactly that—they are mathematical "aliases" that allow us to swap one expression for another to make solving problems much easier.
In this chapter, we will learn how to rewrite trig functions in different ways. This is a vital skill for solving complex equations and is the "secret sauce" for high-level calculus later in the course. Don’t worry if it looks like a lot of formulas at first; we’ll break them down into patterns that are easy to remember!
1. The Foundation: Basic Identities
Before we learn the new A Level tools, let's quickly review the two "Golden Rules" you likely met at AS Level. These are the building blocks for everything else.
Identity 1: The Tangent Rule
\(\tan \theta \equiv \frac{\sin \theta}{\cos \theta}\)
Identity 2: The Pythagorean Identity
\(\sin^2 \theta + \cos^2 \theta \equiv 1\)
Think of it this way: Identity 2 is just Pythagoras' Theorem (\(a^2 + b^2 = c^2\)) hiding inside a circle with a radius of 1!
Quick Review Box:
- You can rearrange Identity 2! For example: \(\sin^2 \theta \equiv 1 - \cos^2 \theta\).
- Common Mistake: Remember that \(\sin^2 \theta\) means \((\sin \theta)^2\). It is not the same as \(\sin \theta^2\)!
2. The New Kids: Reciprocal Identities
At A Level, we introduce three new functions. They are just the "flipped" (reciprocal) versions of the ones you already know.
- Secant: \(\sec \theta \equiv \frac{1}{\cos \theta}\)
- Cosecant: \(\text{cosec } \theta \equiv \frac{1}{\sin \theta}\)
- Cotangent: \(\cot \theta \equiv \frac{1}{\tan \theta} \equiv \frac{\cos \theta}{\sin \theta}\)
Memory Aid: The Third Letter Trick
Struggling to remember which one is which? Just look at the third letter of the new function:
- sec \(\theta\) goes with cos \(\theta\)
- cosec \(\theta\) goes with sin \(\theta\)
- cot \(\theta\) goes with tan \(\theta\)
Linking them together
By dividing the original \(\sin^2 \theta + \cos^2 \theta \equiv 1\) by \(\cos^2 \theta\) or \(\sin^2 \theta\), we get two brand new "Square Identities":
Identity 3: \(1 + \tan^2 \theta \equiv \sec^2 \theta\)
Identity 4: \(1 + \cot^2 \theta \equiv \text{cosec}^2 \theta\)
Example: If an equation has both \(\tan^2 \theta\) and \(\sec \theta\), use Identity 3 to turn everything into \(\sec \theta\) so you can solve it like a quadratic!
Key Takeaway: Use these when you see squared trig terms in an equation and want to "match" them to other terms.
3. Compound Angle Formulae (Addition Rules)
What if you need to find the sine of two angles added together, like \(\sin(A + B)\)? Warning: It is NOT \(\sin A + \sin B\)!
Instead, we use these standard patterns:
- \(\sin(A \pm B) \equiv \sin A \cos B \pm \cos A \sin B\)
- \(\cos(A \pm B) \equiv \cos A \cos B \mp \sin A \sin B\) (Notice the sign flips for cosine!)
- \(\tan(A \pm B) \equiv \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B}\)
Did you know? These allow us to find exact values for angles we don't know. For example, to find \(\sin(75^\circ)\), you can use \(\sin(45^\circ + 30^\circ)\) and plug them into the formula!
Key Takeaway: If you see two different angles inside a trig function, reach for the compound angle formulae.
4. Double Angle Formulae
These are a special case of the compound angle rules where \(A\) and \(B\) are the same (so \(A + A = 2A\)). These are incredibly common in exams.
Sine Double Angle:
\(\sin 2\theta \equiv 2 \sin \theta \cos \theta\)
Cosine Double Angle (The "Triple Threat"):
This one is special because it has three forms. You can choose the one that fits your problem best!
1. \(\cos 2\theta \equiv \cos^2 \theta - \sin^2 \theta\)
2. \(\cos 2\theta \equiv 2\cos^2 \theta - 1\) (Best if you only have cosine terms)
3. \(\cos 2\theta \equiv 1 - 2\sin^2 \theta\) (Best if you only have sine terms)
Tangent Double Angle:
\(\tan 2\theta \equiv \frac{2\tan \theta}{1 - \tan^2 \theta}\)
Key Takeaway: Use these to "downshift" from \(2\theta\) to \(\theta\) so that all parts of your equation are using the same angle.
5. The R-Formula (Harmonic Form)
Sometimes you’ll see an expression like \(3\sin \theta + 4\cos \theta\). It’s hard to solve because there are two different waves happening at once. The R-formula lets us combine them into one single wave.
We can write \(a \sin \theta \pm b \cos \theta\) as either:
\(R \sin(\theta \pm \alpha)\) or \(R \cos(\theta \mp \alpha)\)
Step-by-Step Process:
- Find R: Use Pythagoras! \(R = \sqrt{a^2 + b^2}\).
- Find \(\alpha\): Use \(\tan \alpha = \frac{b}{a}\) (always use the positive values of \(a\) and \(b\)).
- Rewrite: Put them back into the single sine or cosine bracket.
Real-World Analogy: Imagine two people pushing a swing at slightly different times. The swing doesn't move in two different ways; it just follows one single, combined rhythm. The R-formula finds that "combined rhythm."
Bonus Tip: Once in this form, the maximum value is simply \(R\) and the minimum value is \(-R\). This is great for "modeling" questions like tides or Ferris wheels!
6. Top Tips for Proving Identities
Exam questions often ask you to "Show that [Left Side] \(\equiv\) [Right Side]". Don't worry if this seems tricky at first—it’s like a puzzle! Here is your strategy guide:
- Start with the "messier" side: It’s usually easier to simplify a big expression than to expand a small one.
- Change everything to Sine and Cosine: If you see \(\tan, \sec, \text{cosec}\), or \(\cot\), swap them for \(\sin\) and \(\cos\). Often, things will cancel out beautifully.
- Look for squares: If you see a \(\sin^2 \theta\), think about swapping it for \(1 - \cos^2 \theta\).
- Common denominators: If you are adding fractions, get a common denominator first.
- Don't give up! If you get stuck, draw a line and try working backward from the other side. If they meet in the middle, you’ve done it!
Key Takeaway Summary:
- Reciprocals: \(\sec, \text{cosec}, \cot\).
- Squares: \(1 + \tan^2 = \sec^2\) and \(1 + \cot^2 = \text{cosec}^2\).
- Compound/Double Angles: Use these to change the "speed" or combination of angles.
- R-Formula: Use this to turn two waves into one.