Introduction to Trigonometric Identities
Welcome to one of the most useful chapters in Further Pure Mathematics! You might have already met trigonometric equations, but trigonometric identities are slightly different. While an equation is only true for certain values of \( x \), an identity is a mathematical statement that is true for every possible value of the angle.
Think of an identity like a nickname. Whether you call someone "Robert" or "Bob," you are talking about the same person. In the same way, whether you write \( \tan \theta \) or \( \frac{\sin \theta}{\cos \theta} \), you are describing the exact same mathematical value. Master these identities, and you’ll be able to simplify complex expressions and solve difficult equations with ease!
1. The Two Fundamental Identities
There are two primary identities that you must be able to use fluently. These are the "bread and butter" of trigonometry.
The Tangent Identity
This identity relates the three basic ratios. It is provided on your formula sheet, so you don't need to stress about forgetting it, but you will use it so often it will soon become second nature.
\( \tan \theta = \frac{\sin \theta}{\cos \theta} \)
Quick Tip: This is incredibly helpful when you have an equation containing \( \sin \theta \), \( \cos \theta \), and \( \tan \theta \). Converting everything to sine and cosine often makes the path to the solution much clearer!
The Pythagorean Identity
This is perhaps the most famous identity in trigonometry. It comes directly from Pythagoras' Theorem applied to a unit circle.
\( \cos^2 \theta + \sin^2 \theta = 1 \)
Important Notation: Remember that \( \sin^2 \theta \) is just a shorthand way of writing \( (\sin \theta)^2 \). It does not mean \( \sin(\theta^2) \).
You must also be comfortable with this identity in its rearranged forms:
- \( \sin^2 \theta = 1 - \cos^2 \theta \)
- \( \cos^2 \theta = 1 - \sin^2 \theta \)
Analogy: Think of \( \sin^2 \theta \) and \( \cos^2 \theta \) as two pieces of a puzzle that always fit together to make exactly \( 1 \). If you have one piece, you can always find the other.
Key Takeaway: Use these identities to "swap" between different trigonometric terms to make an expression easier to handle.
2. The Addition Formulae (Compound Angles)
Sometimes we need to deal with the sine or cosine of two angles added together, like \( \sin(A + B) \). It is a very common mistake to think that \( \sin(A + B) = \sin A + \sin B \)—this is not true!
Instead, we use the Addition Formulae. These are all provided on your formula sheet in the exam. You do not need to prove them, just know how to apply them.
Sine Addition and Subtraction
\( \sin(A + B) = \sin A \cos B + \cos A \sin B \)
\( \sin(A - B) = \sin A \cos B - \cos A \sin B \)
Cosine Addition and Subtraction
Watch out! Notice that for cosine, the sign in the middle swaps.
\( \cos(A + B) = \cos A \cos B - \sin A \sin B \)
\( \cos(A - B) = \cos A \cos B + \sin A \sin B \)
Tangent Addition and Subtraction
\( \tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \)
\( \tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B} \)
Key Takeaway: Always keep your formula sheet handy when practicing. The most common error is a simple sign mistake (plus instead of minus) in the cosine expansion!
3. Double Angle Formulae
While "Double Angle" formulae aren't listed as a separate section in the syllabus, they are simply a special case of the Addition Formulae where \( B = A \). These are vital for solving equations involving terms like \( \sin 2\theta \) or \( \cos 2\theta \).
Double Angle for Sine
Using \( \sin(A + A) \), we get:
\( \sin 2A = 2 \sin A \cos A \)
Double Angle for Cosine
This one is unique because it can be written in three different ways (by using \( \cos^2 A + \sin^2 A = 1 \) to swap terms):
1. \( \cos 2A = \cos^2 A - \sin^2 A \)
2. \( \cos 2A = 2 \cos^2 A - 1 \) (Useful if you only want cosines)
3. \( \cos 2A = 1 - 2 \sin^2 A \) (Useful if you only want sines)
Did you know? The double angle formula for cosine is often the "secret key" to solving quadratic-style trigonometric equations. If you see a \( \cos 2x \) and a \( \cos x \) in the same equation, use version 2 above to turn everything into \( \cos x \)!
4. Strategies for Proving Identities
Exam questions often ask you to "Show that..." or "Prove that..." one expression equals another. Don't worry if this seems tricky at first; it's a skill that improves with practice.
Step-by-Step Approach:
- Start with the more complicated side: It is usually much easier to simplify a "messy" expression than it is to build up a simple one.
- Convert to Sines and Cosines: If the expression has \( \tan \theta \), use \( \frac{\sin \theta}{\cos \theta} \) to rewrite it.
- Look for Squares: If you see \( \sin^2 \theta \) or \( \cos^2 \theta \), think about the Pythagoras identity (\( \cos^2 \theta + \sin^2 \theta = 1 \)).
- Use Algebraic Skills: Sometimes you need to expand brackets, factorise, or find a common denominator for fractions.
- Keep the destination in mind: Always look at the other side of the identity you are trying to reach. It will give you a hint of what to eliminate.
Key Takeaway: In a "Prove" question, never move terms from one side to the other like an equation. Work on one side only until it looks exactly like the other side.
5. Common Mistakes to Avoid
- Wrong: \( (\sin A + \sin B)^2 = \sin^2 A + \sin^2 B \).
Right: Use FOIL/Grid method! It should be \( \sin^2 A + 2\sin A \sin B + \sin^2 B \). - Wrong: \( \cos(2\theta) = 2\cos\theta \).
Right: You must use the Double Angle formula. You cannot simply "pull the 2 out" of the function. - Confusion with Radian Measure: Ensure your calculator is in the correct mode (Degrees or Radians) depending on the interval given in the question. (See the chapter on Radian Measure for a refresher).
Quick Review Box
Formula Sheet Check: \( \tan \theta = \frac{\sin \theta}{\cos \theta} \) and Addition Formulae are provided.
Must Remember: \( \sin^2 \theta + \cos^2 \theta = 1 \).
Double Angle Hack: \( \sin 2\theta = 2\sin\theta\cos\theta \).
Goal: Identities help turn "unsolvable" equations into simple ones by making all the terms "match."