Welcome to Trigonometry: Pure Mathematics 2!
Hello future mathematician! This chapter, Trigonometry, builds directly on the foundation you established in Pure Mathematics 1. We are moving beyond simple right-angled triangles and delving into how trigonometric functions can model curves, oscillations, and cyclical real-world phenomena.
In P2, we introduce new ways to measure angles (Radians) and meet three powerful new trigonometric functions (Secant, Cosecant, and Cotangent). Don't worry if this sounds intimidating – we will break down every concept step-by-step. Mastering these topics is essential for advanced calculus and modeling!
Section 1: Angles, Arcs, and Area – The Radian Revolution
In P1, you measured angles in degrees. In P2, we primarily use Radians. Why the change? Radians are based on the geometry of the circle itself, making them much more natural and useful for calculus.
What is a Radian?
Imagine a circle. One radian is the angle subtended at the center of a circle when the arc length is exactly equal to the length of the radius.
Did you know? Because the circumference of a circle is \(2\pi r\), a full rotation of \(360^\circ\) is exactly \(2\pi\) radians.
Conversion Rules (The Bridge Between Units)
You must be able to switch easily between degrees and radians:
- To convert Degrees to Radians: Multiply by \(\frac{\pi}{180}\).
- To convert Radians to Degrees: Multiply by \(\frac{180}{\pi}\).
Memory Aid: When converting *to* radians, the \(\pi\) must be on top!
Key Equivalents:
- \(360^\circ = 2\pi\) rad
- \(180^\circ = \pi\) rad
- \(90^\circ = \frac{\pi}{2}\) rad
- \(30^\circ = \frac{\pi}{6}\) rad
Arc Length and Sector Area Formulas (Using Radians)
These formulas are crucial and only work if the angle \(\theta\) is measured in radians.
1. Arc Length (\(L\))
The length of the curved edge of a sector (like the crust of a pizza slice).
$$L = r\theta$$
Where: \(r\) is the radius and \(\theta\) is the angle in radians.
2. Sector Area (\(A\))
The total area of the pizza slice.
$$A = \frac{1}{2}r^2\theta$$
If you are given the arc length \(L\), you can also calculate the area using the alternative formula:
$$A = \frac{1}{2}rL$$
Step-by-Step Example: Finding Arc Length
Question: A sector has a radius of 6 cm and an angle of \(120^\circ\). Find the arc length.
Step 1: Convert to Radians.
$$\theta = 120^\circ \times \frac{\pi}{180} = \frac{2\pi}{3} \text{ rad}$$
Step 2: Apply the Formula.
$$L = r\theta = 6 \times \frac{2\pi}{3} = 4\pi \text{ cm}$$
1. P2 uses radians because they are geometrically natural.
2. Always check the required angle unit before using \(L = r\theta\) or \(A = \frac{1}{2}r^2\theta\).
Section 2: The Reciprocal Crew – Secant, Cosecant, and Cotangent
In P1, you used sine, cosine, and tangent. In P2, we introduce their reciprocals. These new functions are defined simply as one divided by the original function.
Definitions of the New Trigonometric Functions
These definitions are absolutely fundamental and must be memorized:
1. Secant (\(\sec \theta\)): The reciprocal of cosine.
$$ \sec \theta = \frac{1}{\cos \theta} $$2. Cosecant (\(\csc \theta\) or sometimes \(\text{cosec } \theta\)): The reciprocal of sine.
$$ \csc \theta = \frac{1}{\sin \theta} $$3. Cotangent (\(\cot \theta\)): The reciprocal of tangent.
$$ \cot \theta = \frac{1}{\tan \theta} $$Since \(\tan \theta = \frac{\sin \theta}{\cos \theta}\), the cotangent can also be written as:
$$ \cot \theta = \frac{\cos \theta}{\sin \theta} $$Memory Aid: The Third Letter Trick
A simple trick to avoid mixing them up:
- Secant starts with 's', but its third letter is 'c' (for Cosine).
- Csc (Cosecant) starts with 'c', but its third letter is 's' (for Sine).
- Cotangent starts with 'c', and its reciprocal, Tangent, starts with 't'.
Graphs of Reciprocal Functions
Since these functions are reciprocals, they have vertical asymptotes wherever the original function is zero (because dividing by zero is undefined).
- \(\sec \theta\) has asymptotes when \(\cos \theta = 0\), i.e., at \(90^\circ, 270^\circ, \dots\).
- \(\csc \theta\) has asymptotes when \(\sin \theta = 0\), i.e., at \(0^\circ, 180^\circ, 360^\circ, \dots\).
Students sometimes confuse \(\sec \theta\) with \(\frac{1}{\sin \theta}\) because 'S' and 'C' seem naturally paired. Remember the third letter trick! If you see \(\sec\), think \(\cos\).
Section 3: Essential Trigonometric Identities (The P2 Power Tools)
The identities you learned in P1 (\(\sin^2\theta + \cos^2\theta = 1\)) are the basis for the two powerful new identities you need for P2.
Deriving the New Identities
We start with the fundamental identity: $$ \sin^2\theta + \cos^2\theta = 1 $$
Identity A: The Tan/Sec Identity
We take the fundamental identity and divide every term by \(\cos^2\theta\):
$$ \frac{\sin^2\theta}{\cos^2\theta} + \frac{\cos^2\theta}{\cos^2\theta} = \frac{1}{\cos^2\theta} $$
Using the definitions of tangent and secant, this simplifies to:
$$ 1 + \tan^2\theta = \sec^2\theta $$
(Memorize this form!)
Identity B: The Cot/Csc Identity
Now, we take the fundamental identity and divide every term by \(\sin^2\theta\):
$$ \frac{\sin^2\theta}{\sin^2\theta} + \frac{\cos^2\theta}{\sin^2\theta} = \frac{1}{\sin^2\theta} $$
Using the definitions of cotangent and cosecant, this simplifies to:
$$ 1 + \cot^2\theta = \csc^2\theta $$
(Memorize this form!)
Summary of P2 Identities
- \( \sin^2\theta + \cos^2\theta = 1 \)
- \( 1 + \tan^2\theta = \sec^2\theta \) (The 'Tan and Sec' family)
- \( 1 + \cot^2\theta = \csc^2\theta \) (The 'Cot and Csc' family)
Key Takeaway for Proofs: When asked to prove an identity (e.g., show that \(\frac{1+\sec\theta}{\sec\theta} \equiv 1 + \cos\theta\)), always try to convert everything back to its basic components (\(\sin\theta\) and \(\cos\theta\)) first. Use the LHS (Left-Hand Side) and simplify it until it matches the RHS (Right-Hand Side).
Example: Proving an Identity
We want to simplify \((\sec\theta - 1)(\sec\theta + 1)\).
$$ (\sec\theta - 1)(\sec\theta + 1) = \sec^2\theta - 1^2 $$
Using the identity \(1 + \tan^2\theta = \sec^2\theta\), we can rearrange it to get \(\tan^2\theta = \sec^2\theta - 1\).
Therefore, the expression simplifies directly to \(\tan^2\theta\).
Section 4: Solving Advanced Trigonometric Equations
In P2, you will solve equations involving the new reciprocal functions and the new identities. The general strategy is always the same: convert back to sin, cos, or tan whenever possible.
Strategy for Reciprocal Equations
If you have an equation like \(\sec x = 3\), follow these steps:
Step 1: Isolate the Reciprocal Function
Make the reciprocal function the subject (e.g., \(\sec x = 3\)).
Step 2: Flip it!
Convert the equation back to its basic function by taking the reciprocal of both sides.
If \(\sec x = 3\), then \(\cos x = \frac{1}{3}\).
Step 3: Solve Using CAST and Quadrants (The P1 Method)
Use the inverse function (e.g., \(x = \cos^{-1}(\frac{1}{3})\)) to find the principal value (the first angle, usually in the range \(0^\circ < x < 90^\circ\)). Then, use the CAST diagram and the given domain to find all possible solutions.
Encouragement: Once you flip the equation, the solving process is exactly what you mastered in P1!
Example: Solving a Cosecant Equation
Solve \(\csc x = -2\) for \(0^\circ \leq x < 360^\circ\).
Step 1 & 2: Flip it!
$$\frac{1}{\sin x} = -2 \implies \sin x = -\frac{1}{2}$$
Step 3: Find Principal Value.
We ignore the negative sign to find the reference angle (\(\alpha\)):
$$\alpha = \sin^{-1}(\frac{1}{2}) = 30^\circ$$
Step 4: Use CAST.
Since \(\sin x\) is negative, solutions are in the Third and C fourth quadrants.
- Quadrant III: \(x = 180^\circ + 30^\circ = 210^\circ\)
- Quadrant IV: \(x = 360^\circ - 30^\circ = 330^\circ\)
Solving Equations Involving Identities (Quadratic Form)
Many P2 equations require you to use the identities to reduce the equation into a single function (sin, cos, or tan).
Example: Solve \(2\sec^2\theta + \tan\theta = 4\).
The Challenge: This equation has two different functions: \(\sec^2\theta\) and \(\tan\theta\).
Step 1: Convert to a Single Function.
Use the identity \( \sec^2\theta = 1 + \tan^2\theta \). Substitute this into the equation:
Step 2: Rearrange into a Quadratic Equation.
$$ 2 + 2\tan^2\theta + \tan\theta = 4 $$
$$ 2\tan^2\theta + \tan\theta - 2 = 0 $$
Step 3: Solve the Quadratic.
Let \(y = \tan\theta\). Solve \(2y^2 + y - 2 = 0\) using the quadratic formula (since it doesn't factor easily).
$$ y = \frac{-1 \pm \sqrt{1^2 - 4(2)(-2)}}{4} = \frac{-1 \pm \sqrt{17}}{4} $$
Step 4: Solve for \(\theta\).
You will now solve two separate equations: \(\tan\theta = \frac{-1 + \sqrt{17}}{4}\) and \(\tan\theta = \frac{-1 - \sqrt{17}}{4}\). You find the principal values and use the appropriate CAST quadrants (depending on whether the tangent result is positive or negative).
Always aim for simplicity. If you see a reciprocal function, flip it. If you see mixed functions (\(\sec\) and \(\tan\)), use the identities to switch to a single function (usually creating a quadratic equation).