Unit P3: Pure Mathematics 3 - Comprehensive Study Notes on Trigonometry
Welcome to the exciting world of P3 Trigonometry! If you thought P1 and P2 covered everything about triangles, think again. P3 takes trigonometric functions and turns them into powerful algebraic tools.
This chapter is fundamental for success in later P3 topics, especially differentiation, integration, and algebraic manipulation. Don't worry if this seems tricky at first; we will break down the complex identities into simple, memorable rules. By the end, you'll be able to solve complex equations that were previously impossible!
1. The Power of Combination: Compound Angle Identities
The Compound Angle Identities allow us to expand trigonometric functions of summed or subtracted angles, like \(\sin(A+B)\) or \(\cos(x-45^{\circ})\). These are the foundation for nearly everything else in P3 Trigonometry.
1.1 The Six Core Compound Identities
You need to know these six identities by heart, but here are some tricks to help you remember which functions and signs go where:
Sine Identities (The "Sharing" Functions):
- \(\sin(A + B) \equiv \sin A \cos B + \cos A \sin B\)
- \(\sin(A - B) \equiv \sin A \cos B - \cos A \sin B\)
Memory Aid: Sine is friendly—it shares the function (Sine, Cosine, Cosine, Sine) and the signs match (plus stays plus, minus stays minus).
Cosine Identities (The "Keeping" Functions):
- \(\cos(A + B) \equiv \cos A \cos B - \sin A \sin B\)
- \(\cos(A - B) \equiv \cos A \cos B + \sin A \sin B\)
Memory Aid: Cosine is self-centred—it keeps the same functions together (Cosine, Cosine, Sine, Sine) and the signs swap (plus becomes minus, minus becomes plus).
Tangent Identities:
- \(\tan(A + B) \equiv \frac{\tan A + \tan B}{1 - \tan A \tan B}\)
- \(\tan(A - B) \equiv \frac{\tan A - \tan B}{1 + \tan A \tan B}\)
Tip: The sign in the numerator matches the sign of the angle combination, and the sign in the denominator is the opposite.
Key Takeaway: Compound angles are the algebraic way to simplify functions of combined angles. Use the mnemonics to avoid mixing up the signs!
2. Double the Trouble: Double Angle Identities
The Double Angle Identities are simply special cases of the compound identities where \(A = B\). They are essential for manipulating expressions, especially when solving equations or preparing for integration.
2.1 Deriving the Double Angle Formulas
Setting \(B = A\) in the compound angle formula for \(\sin(A+B)\):
\(\sin(A + A) = \sin A \cos A + \cos A \sin A\)
The Sine Double Angle:
\(\sin(2A) \equiv 2\sin A \cos A\)
Setting \(B = A\) in the compound angle formula for \(\cos(A+B)\):
\(\cos(A + A) = \cos A \cos A - \sin A \sin A\)
The Primary Cosine Double Angle:
\(\cos(2A) \equiv \cos^2 A - \sin^2 A\)
2.2 The Three Forms of \(\cos(2A)\) - CRITICAL!
Using the Pythagorean Identity \(\sin^2 A + \cos^2 A \equiv 1\), we can substitute and create two other extremely useful forms of \(\cos(2A)\). These are used constantly to reduce the power of trig functions from 2 down to 1 (linearisation).
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In terms of only \(\cos^2 A\):
Substitute \(\sin^2 A = 1 - \cos^2 A\):
\(\cos(2A) \equiv \cos^2 A - (1 - \cos^2 A)\)
\(\cos(2A) \equiv 2\cos^2 A - 1\) -
In terms of only \(\sin^2 A\):
Substitute \(\cos^2 A = 1 - \sin^2 A\):
\(\cos(2A) \equiv (1 - \sin^2 A) - \sin^2 A\)
\(\cos(2A) \equiv 1 - 2\sin^2 A\)
The Tangent Double Angle:
\(\tan(2A) \equiv \frac{2\tan A}{1 - \tan^2 A}\)
Did you know? These formulas aren't just for \(2A\). If you see \(\sin(4x)\), you can write it as \(2\sin(2x)\cos(2x)\). If you see \(\cos(x)\), you can write it as \(2\cos^2(x/2) - 1\). The angle on the left is always double the angle on the right.
Common Mistake to Avoid: Confusing the three cosine double angle identities. If you need to substitute for \(\cos^2 A\), rearrange the relevant formula:
\(\cos^2 A \equiv \frac{1}{2}(1 + \cos(2A))\) (Extremely useful for integration!)
Key Takeaway: Double Angle identities simplify functions of \(2A\). The three forms of \(\cos(2A)\) are crucial for changing the power of a trigonometric term.
3. The R-Formula: Simplifying \(a\cos\theta + b\sin\theta\)
Imagine you have two simple sine waves, one slightly shifted from the other, added together. They create a new, messy wave. The R-Formula (or Auxiliary Angle Form) is a tool that allows us to combine \(a\cos\theta + b\sin\theta\) into a single, clean trigonometric function, usually in the form \(R\cos(\theta \pm \alpha)\) or \(R\sin(\theta \pm \alpha)\).
3.1 Why Use the R-Formula? (The Analogy)
Think of \(y = 3\sin\theta + 4\cos\theta\).
We can't easily find the maximum value or solve the equation \(3\sin\theta + 4\cos\theta = 2\) because we have two different functions. The R-formula converts this complex expression into a single, shifted sine or cosine curve, making solving easy!
3.2 Step-by-Step Method (Using \(R\cos(\theta - \alpha)\))
Let's express \(a\cos\theta + b\sin\theta\) in the form \(R\cos(\theta - \alpha)\), where \(R > 0\) and \(0^{\circ} < \alpha < 90^{\circ}\).
Step 1: Expand the Target Form
Use the Compound Angle Identity for Cosine (Signs Swap!):
\(R\cos(\theta - \alpha) \equiv R(\cos\theta \cos\alpha + \sin\theta \sin\alpha)\)
\(R\cos(\theta - \alpha) \equiv (R\cos\alpha)\cos\theta + (R\sin\alpha)\sin\theta\)
Step 2: Compare Coefficients
We equate this expansion to the original expression \(a\cos\theta + b\sin\theta\):
(1) \(a = R\cos\alpha\) (Coefficient of \(\cos\theta\))
(2) \(b = R\sin\alpha\) (Coefficient of \(\sin\theta\))
Step 3: Calculate R (The Amplitude)
Square and add equations (1) and (2):
\(a^2 + b^2 = R^2\cos^2\alpha + R^2\sin^2\alpha\)
\(a^2 + b^2 = R^2(\cos^2\alpha + \sin^2\alpha)\)
Since \(\cos^2\alpha + \sin^2\alpha = 1\):
\(R = \sqrt{a^2 + b^2}\)
Step 4: Calculate \(\alpha\) (The Phase Shift)
Divide equation (2) by equation (1):
\(\frac{R\sin\alpha}{R\cos\alpha} = \frac{b}{a}\)
\(\tan\alpha = \frac{b}{a}\)
\(\alpha = \arctan\left(\frac{b}{a}\right)\)
Crucial Note on Signs:
Always calculate \(R\) and \(\alpha\) using the positive values of \(a\) and \(b\). The choice of the target form (e.g., \(R\sin(\theta + \alpha)\) vs \(R\cos(\theta - \alpha)\)) already handles the signs from the original expression.
Applications of the R-Formula:
- Finding Max/Min Values: The maximum value of \(R\cos(\theta \pm \alpha)\) is always \(R\), and the minimum value is \(-R\).
- Solving Equations: It reduces equations like \(a\cos\theta + b\sin\theta = k\) into the solvable form \(R\cos(\theta - \alpha) = k\).
Quick Review:
R is the hypotenuse of a right-angled triangle with sides a and b. \(\alpha\) is the angle.
Key Takeaway: The R-formula is a technique used to combine a linear combination of sine and cosine functions into a single, shifted trigonometric function \(R f(\theta \pm \alpha)\), simplifying maximum/minimum problems and equation solving.
4. Solving Trigonometric Equations in P3
In P3, solving equations involves combining all the P2 techniques (CAST diagrams, period/domain checks) with the P3 identities.
4.1 Strategy for Using Identities to Solve Equations
When solving equations, your main goal is to reduce the equation so it contains only one trigonometric function and only one angle term (e.g., only \(\sin x\), not \(\sin x\) and \(\cos x\), or \(\sin 2x\)).
- Check the Function Type: If you have a mixture (e.g., \(\sin 2x\) and \(\cos x\)), use the Double Angle Identity to convert the multiple angle into the single angle: \(\sin 2x = 2\sin x \cos x\).
- Check the Angle Type: If you have an expression like \(3\sin x + 4\cos x = 1\), use the R-Formula to combine them into one function: \(5\cos(x - 36.9^{\circ}) = 1\).
- Check the Power Type: If you have \(\cos 2x = 3\cos x - 1\), substitute the appropriate form of \(\cos 2x\) to eliminate the multiple angle and create a quadratic in \(\cos x\). (Use \(\cos 2x = 2\cos^2 x - 1\)).
- Solve the Resulting Simple or Quadratic Equation: Find the principal value using the inverse function.
- Find All Solutions: Use the CAST diagram or graph symmetry within the given domain.
Example Process: Solve \(\cos 2\theta + \sin\theta = 0\) for \(0^{\circ} \le \theta < 360^{\circ}\).
Step 1: Simplify. The equation has a mix of \(2\theta\) and \(\theta\). Use the \(\cos 2\theta\) identity that results in only \(\sin\theta\) terms.
\((1 - 2\sin^2\theta) + \sin\theta = 0\)
Step 2: Rearrange to Quadratic.
\(2\sin^2\theta - \sin\theta - 1 = 0\)
Step 3: Solve the Quadratic. Let \(S = \sin\theta\).
\((2S + 1)(S - 1) = 0\)
Therefore, \(\sin\theta = 1\) or \(\sin\theta = -1/2\).
Step 4: Find Final Solutions.
If \(\sin\theta = 1\), then \(\theta = 90^{\circ}\).
If \(\sin\theta = -1/2\), principal value is \(-30^{\circ}\). Solutions in the range (using C and T quadrants) are \(180^{\circ} + 30^{\circ} = 210^{\circ}\) and \(360^{\circ} - 30^{\circ} = 330^{\circ}\).
Solutions: \(90^{\circ}, 210^{\circ}, 330^{\circ}\).
5. Inverse Trigonometric Functions (\(\arcsin, \arccos, \arctan\))
We often deal with inverse trig functions (also called arc functions), like \(\arcsin x\), which tell us the angle whose sine is \(x\).
5.1 The Need for Restricted Domains (Principal Values)
For a function to have an inverse, it must be one-to-one (the inverse must pass the vertical line test). Since standard trigonometric functions are periodic (many angles give the same value), we must restrict their domains to ensure the inverse is a true function. The output of an inverse function is called the principal value.
Quick Definition:
- \(\mathbf{y = \arcsin x}\) (or \(\sin^{-1} x\)): means \(x = \sin y\).
- \(\mathbf{y = \arccos x}\) (or \(\cos^{-1} x\)): means \(x = \cos y\).
- \(\mathbf{y = \arctan x}\) (or \(\tan^{-1} x\)): means \(x = \tan y\).
5.2 Domains and Ranges (The Principal Values)
These restrictions are essential. They define the specific angle output your calculator will give you.
The Inverse Sine Function: \(y = \arcsin x\)
- Domain (x-values allowed): \(-1 \le x \le 1\)
- Range (y-values output): \(-\frac{\pi}{2} \le y \le \frac{\pi}{2}\) (or \(-90^{\circ} \le y \le 90^{\circ}\))
The Inverse Cosine Function: \(y = \arccos x\)
- Domain (x-values allowed): \(-1 \le x \le 1\)
- Range (y-values output): \(0 \le y \le \pi\) (or \(0^{\circ} \le y \le 180^{\circ}\))
The Inverse Tangent Function: \(y = \arctan x\)
- Domain (x-values allowed): \(x \in \mathbb{R}\) (All real numbers)
- Range (y-values output): \(-\frac{\pi}{2} < y < \frac{\pi}{2}\) (or \(-90^{\circ} < y < 90^{\circ}\))
Tip for Struggling Students: When you are asked to evaluate an inverse function, always check which quadrant the restricted range falls into. \(\arcsin\) and \(\arctan\) use Quadrants I and IV. \(\arccos\) uses Quadrants I and II.
Key Takeaway: Inverse trig functions only output angles within a specific, restricted range (the principal values) because the original trig functions must be one-to-one.
Congratulations! You have mastered the core algebraic tools of P3 Trigonometry. Practice applying these identities to complex equations, and you will find they simplify quickly. Keep up the hard work!