Introduction to Trigonometric Identities

Welcome to one of the most useful chapters in IB Mathematics! Think of trigonometric identities as "mathematical synonyms." Just as "happy" and "joyful" mean the same thing in English, these identities allow us to rewrite complex trigonometric expressions in simpler, more manageable forms. Whether you are aiming for SL or HL, mastering these tools is the secret to solving tricky equations and simplifying calculus problems later on.

In this chapter, we focus on the fundamental Pythagorean identity and the double angle formulae. Don't worry if these look like a lot of symbols at first—once you see how they are connected, they become much easier to remember!

1. The Fundamental Pythagorean Identity

The most important identity in all of trigonometry comes directly from the Unit Circle and the Pythagorean Theorem (\(a^2 + b^2 = c^2\)).

The Identity:

\(\cos^2 \theta + \sin^2 \theta = 1\)

Why does this work?

On a unit circle, any point has coordinates \((\cos \theta, \sin \theta)\). Since the radius of the unit circle is \(1\), the horizontal distance is \(\cos \theta\) and the vertical distance is \(\sin \theta\). Squaring them and adding them must equal the square of the hypotenuse, which is \(1^2 = 1\).

Useful Variations:

You can rearrange this identity to help solve for one ratio in terms of another:

  • \(\cos^2 \theta = 1 - \sin^2 \theta\)
  • \(\sin^2 \theta = 1 - \cos^2 \theta\)

Quick Tip: Remember that \(\sin^2 \theta\) is just a shorthand way of writing \((\sin \theta)^2\). It does not mean \(\sin(\theta^2)\)!

Common Mistake to Avoid: A very common error is thinking that \(\cos \theta + \sin \theta = 1\). This is false! The identity only works when the terms are squared.


2. The Relationship Between Ratios

As you learned in previous chapters, the tangent of an angle is simply the ratio of sine to cosine:

\(\tan \theta = \frac{\sin \theta}{\cos \theta}\)

Did you know? This relationship allows us to turn any equation involving \(\sin\), \(\cos\), and \(\tan\) into an equation with just \(\sin\) and \(\cos\), making it much easier to simplify.

Key Takeaway: If you are stuck on a proof or an equation, try converting everything into \(\sin \theta\) and \(\cos \theta\) first!


3. Double Angle Formulae (SL & HL)

Double angle formulae allow us to express the trig ratio of twice an angle (\(2\theta\)) in terms of the original angle (\(\theta\)). These are essential for solving equations where the angles don't match (e.g., an equation containing both \(\sin 2x\) and \(\sin x\)).

A. Double Angle for Sine

\(\sin 2\theta = 2\sin \theta \cos \theta\)

Example: If you see \(\sin 10x\), you can rewrite it as \(2\sin 5x \cos 5x\).

B. Double Angle for Cosine

The cosine double angle is unique because it has three different forms. All of them are correct, but usually one is more "helpful" than the others depending on your goal.

  1. \(\cos 2\theta = \cos^2 \theta - \sin^2 \theta\)
  2. \(\cos 2\theta = 2\cos^2 \theta - 1\) (Best if you want only cosine in your equation)
  3. \(\cos 2\theta = 1 - 2\sin^2 \theta\) (Best if you want only sine in your equation)

How to choose? If you are solving an equation like \(\cos 2x + \sin x = 0\), you should pick the third version (\(1 - 2\sin^2 x\)) so that the entire equation is written using only \(\sin x\). This turns it into a quadratic equation that you can factor!


4. Double Angle for Tangent (HL Only)

If you are a Higher Level student, you also need to know the double angle formula for tangent, which is derived from the compound angle identities:

\(\tan 2\theta = \frac{2\tan \theta}{1 - \tan^2 \theta}\)

Quick Review Box:

  • Sine Double Angle: Always results in a product (\(2 \cdot \sin \cdot \cos\)).
  • Cosine Double Angle: Always results in a sum or difference involving squares.
  • Tangent Double Angle: Results in a fraction.

5. Strategy: How to Prove Identities

In the IB exam, you might be asked to "Show that" one side of an equation equals the other. Here is a step-by-step approach for even the toughest proofs:

  1. Start with the more complicated side: It is usually easier to simplify a mess than to "build" a complex expression from a simple one.
  2. Look for Double Angles: If one side has \(2\theta\) and the other has \(\theta\), use your double angle formulae immediately.
  3. Substitute the Pythagorean Identity: If you see a \(1\) and a \(\sin^2 \theta\), try replacing the \(1\) with \(\sin^2 \theta + \cos^2 \theta\) or rearranging it.
  4. Use Algebra: Don't forget your standard algebra skills! Factorizing, expanding brackets, and finding common denominators for fractions are often the missing steps.
  5. Keep the target in mind: Look at the side you aren't working on. If it only has sines, try to eliminate all cosines from your working side.

Common Mistake: Never move terms from one side of the equals sign to the other during a "Show that" proof. Treat the Left Hand Side (LHS) and Right Hand Side (RHS) as two separate walls until they look exactly the same!


Summary Checklist

Before moving to the next chapter on Trigonometric Equations, make sure you can:

  • State the Pythagorean Identity: \(\cos^2 \theta + \sin^2 \theta = 1\).
  • Recall the sine double angle formula.
  • Recognize all three versions of the cosine double angle formula.
  • (HL) Use the tangent double angle formula.
  • Convert \(\tan \theta\) into \(\frac{\sin \theta}{\cos \theta}\) to simplify expressions.

Note: Always keep your IB Formula Booklet handy! While you should memorize these for speed, they are provided in the booklet for you to verify during the exam.