Introduction to Trigonometric Identities

Welcome to one of the most useful chapters in Pure Mathematics! In this section, we are going to learn about Trigonometric Identities. Don't let the name intimidate you—an "identity" is simply a mathematical way of saying that two things are identical, no matter what value you pick for the angle \( \theta \).

Think of identities like mathematical aliases. Just as "Clark Kent" and "Superman" are different names for the same person, \( \tan \theta \) and \( \frac{\sin \theta}{\cos \theta} \) are different names for the same mathematical value. Understanding these connections allows you to simplify incredibly messy equations and solve problems that look impossible at first glance!

Did you know? Trigonometric identities are used in everything from designing the curves of a roller coaster to helping your smartphone GPS calculate your exact location on Earth.


1. The Tangent Identity

The first identity you need to master connects the three main trigonometric ratios: Sine, Cosine, and Tangent. It is defined as:

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

Why is this true?

If you remember your basic SOH CAH TOA from GCSE:

  • \( \sin \theta = \frac{Opposite}{Hypotenuse} \)
  • \( \cos \theta = \frac{Adjacent}{Hypotenuse} \)
  • \( \tan \theta = \frac{Opposite}{Adjacent} \)

If you divide the Sine formula by the Cosine formula, the "Hypotenuse" parts cancel out, leaving you with \( \frac{Opposite}{Adjacent} \), which is exactly what Tangent is!

How to use it:

Whenever you see a \( \tan \theta \) in a difficult equation, you can swap it for \( \frac{\sin \theta}{\cos \theta} \). This is usually the first step in simplifying a problem because it's much easier to work with just two variables (sin and cos) rather than three.

Quick Tip: This identity also works for powers! For example, \( \tan^2 \theta = \frac{\sin^2 \theta}{\cos^2 \theta} \).

Key Takeaway: Tangent is just Sine divided by Cosine. Use this to reduce the number of different trig terms in an equation.


2. The Pythagorean Identity

This is the "superstar" of trigonometric identities. It links Sine and Cosine together in a way that always equals 1:

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

The Analogy: The Unit Circle

Imagine a ladder of length 1 leaning against a wall. The height it reaches is \( \sin \theta \) and the distance from the wall is \( \cos \theta \). Because of Pythagoras' Theorem (\( a^2 + b^2 = c^2 \)), the horizontal distance squared plus the vertical height squared must equal the ladder length squared (\( 1^2 \)). Since \( 1^2 = 1 \), we get our identity!

Rearranging the Identity

You will often need to "camouflage" this identity to solve problems. You can subtract terms from both sides to get these useful versions:

  • To replace \( \sin^2 \theta \), use: \( \sin^2 \theta = 1 - \cos^2 \theta \)
  • To replace \( \cos^2 \theta \), use: \( \cos^2 \theta = 1 - \sin^2 \theta \)

Common Mistake Alert: Make sure you write the square in the right place! \( \sin^2 \theta \) means \( (\sin \theta)^2 \). However, \( \sin \theta^2 \) means you are squaring the angle first, which is completely different!

Key Takeaway: If you have a \( \sin^2 \theta \) and want it to become a Cosine (or vice versa), the Pythagorean identity is your best friend.


3. Solving Trigonometric Equations

The main reason we learn these identities is to solve equations. Often, an exam question will give you an equation with a mixture of Sine, Cosine, or Tangent, and you need to make them "match" before you can solve it.

Step-by-Step Example:

Solve \( 2\sin^2 \theta - \cos \theta - 1 = 0 \) for \( 0^\circ \leq \theta \leq 360^\circ \).

  1. Identify the "mismatch": We have a \( \sin^2 \theta \) and a \( \cos \theta \). We can't solve it like this.
  2. Swap it out: Since we have a \( \cos \theta \) (which isn't squared), it's easier to change the \( \sin^2 \theta \) into Cosines using our identity: \( \sin^2 \theta = 1 - \cos^2 \theta \).
  3. Substitute: Replace the term in the equation: \( 2(1 - \cos^2 \theta) - \cos \theta - 1 = 0 \).
  4. Expand and Simplify: \( 2 - 2\cos^2 \theta - \cos \theta - 1 = 0 \), which becomes \( -2\cos^2 \theta - \cos \theta + 1 = 0 \).
  5. Solve the Quadratic: Multiply by -1 to make it prettier: \( 2\cos^2 \theta + \cos \theta - 1 = 0 \). Now you can treat \( \cos \theta \) like \( x \) and solve using factoring or the quadratic formula!

Don't worry if this seems tricky at first! The goal is always the same: use identities to make every trig term in the equation the same type.


4. Proving Trigonometric Identities

Sometimes, the question will ask you to "Prove" or "Show that" one side of an equation equals the other. This is like a mathematical puzzle.

Top Tips for Proofs:

  • Start with the "messy" side: It is much easier to simplify a complex expression than to make a simple one more complicated.
  • Change everything to Sine and Cosine: If you see a Tangent, use \( \tan \theta = \frac{\sin \theta}{\cos \theta} \) immediately.
  • Look for common denominators: If you have fractions, add them together just like you would in basic algebra.
  • Keep the goal in sight: Always look at the other side of the equation to see what your final answer should look like.

Key Takeaway: In a proof, you are not "solving" for \( \theta \). You are just rearranging one side until it looks exactly like the other side.


Quick Review Box

Memory Aid (Mnemonic):
Sin over Cos is Tan (Silly Cats Talk)
Sin squared plus Cos squared is One (Super Cool One)

  • Identity 1: \( \tan \theta = \frac{\sin \theta}{\cos \theta} \)
  • Identity 2: \( \sin^2 \theta + \cos^2 \theta = 1 \)
  • Goal: Use these to make all terms in an equation the same (e.g., all sines or all cosines).

Common Pitfalls to Avoid

1. Dividing by zero: Be careful when dividing by \( \cos \theta \) to get a Tangent. If \( \cos \theta = 0 \), that division isn't allowed! In most AS level problems, the interval given will help you avoid this, but it's good to keep in mind.

2. Algebraic slips: Many students forget that \( ( \sin \theta + \cos \theta )^2 \) is NOT \( \sin^2 \theta + \cos^2 \theta \). You must expand it like a double bracket: \( \sin^2 \theta + 2\sin \theta\cos \theta + \cos^2 \theta \).

3. Negative signs: When replacing \( \sin^2 \theta \) with \( (1 - \cos^2 \theta) \), always use brackets if there is a number in front of it to avoid sign errors!