Chapter 1.4: Trigonometry (AS 1 Pure Mathematics)
Welcome to your study notes for AS 1 Trigonometry! Trigonometry connects geometric shapes with algebraic functions. In GCSE, you worked mostly with right-angled triangles using basic ratios (\(\text{SOH CAH TOA}\)). At AS Level, we expand these ideas to work with any triangle, explore angles of any size, prove trigonometric identities, and solve multi-step trigonometric equations in degrees.
Don't worry if this seems like a big step up at first! By breaking down each method step-by-step and keeping an eye out for standard exam traps, you will quickly build confidence.
1. Non-Right-Angled Triangles: Sine & Cosine Rules
In any general triangle \(ABC\), side \(a\) is opposite angle \(A\), side \(b\) is opposite angle \(B\), and side \(c\) is opposite angle \(C\).
A. The Sine Rule
Use the Sine Rule when you have a "matching pair" consisting of a side and its opposite angle:
• To find a side:
\(\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\)
• To find an angle:
\(\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}\)
The Ambiguous Case of the Sine Rule
When you are given two sides and a non-included acute angle (Side-Side-Angle or SSA), there may be two valid triangles:
1. The calculator gives you the acute angle: \(\theta_1 = \sin^{-1}(\text{value})\).
2. A second possible obtuse angle exists: \(\theta_2 = 180^\circ - \theta_1\).
3. Check: If \(\theta_2\) plus the given angle is strictly less than \(180^\circ\), then \(\theta_2\) produces a valid second triangle.
B. The Cosine Rule
Use the Cosine Rule when you have three sides (SSS) or two sides and the included angle (SAS):
• To find a missing side:
\(a^2 = b^2 + c^2 - 2bc\cos A\)
• To find a missing angle (rearranged):
\(\cos A = \frac{b^2 + c^2 - a^2}{2bc}\)
Examiner Warning: A frequent algebraic mistake is calculating \(b^2 + c^2 - 2bc\) before multiplying by \(\cos A\). Remember that multiplication takes priority in the order of operations! Always evaluate \(2bc\cos A\) as a single term before subtracting it from \(b^2 + c^2\).
C. Area of a Triangle
The area of any triangle can be calculated if you know two sides and the included angle (the angle between them):
\(\text{Area} = \frac{1}{2}ab\sin C\)
Section 1 Key Takeaway: Choose the Sine Rule when you have an angle-side opposite pair. Choose the Cosine Rule when you have two sides with the angle between them (SAS) or all three sides (SSS).
2. Trigonometric Functions & Exact Values
At AS Level, angles are no longer limited to \(0^\circ\) to \(90^\circ\). We can evaluate trigonometric functions for any angle \(\theta \in \mathbb{R}\) using the coordinate plane and standard quadrants.
A. The CAST Diagram (Quadrant Rule)
The sign (positive or negative) of \(\sin \theta\), \(\cos \theta\), and \(\tan \theta\) depends on the quadrant in which the angle lies:
• Quadrant 1 (\(0^\circ\) to \(90^\circ\)): All three ratios are positive.
• Quadrant 2 (\(90^\circ\) to \(180^\circ\)): Sine only is positive (\(\sin(180^\circ - \theta) = \sin\theta\)).
• Quadrant 3 (\(180^\circ\) to \(270^\circ\)): Tangent only is positive (\(\tan(180^\circ + \theta) = \tan\theta\)).
• Quadrant 4 (\(270^\circ\) to \(360^\circ\)): Cosine only is positive (\(\cos(360^\circ - \theta) = \cos\theta\)).
Memory Trick: Read the quadrants counter-clockwise starting in Quadrant 4 to remember the word CAST (or Quadrant 1 to 4: All Students Take Calculus).
B. Exact Values for Standard Angles
You must know the exact values for the key angles (\(0^\circ, 30^\circ, 45^\circ, 60^\circ, 90^\circ\)):
• For \(\sin \theta\):
\(\sin 0^\circ = 0\)
\(\sin 30^\circ = \frac{1}{2}\)
\(\sin 45^\circ = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}\)
\(\sin 60^\circ = \frac{\sqrt{3}}{2}\)
\(\sin 90^\circ = 1\)
• For \(\cos \theta\):
\(\cos 0^\circ = 1\)
\(\cos 30^\circ = \frac{\sqrt{3}}{2}\)
\(\cos 45^\circ = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}\)
\(\cos 60^\circ = \frac{1}{2}\)
\(\cos 90^\circ = 0\)
• For \(\tan \theta\):
\(\tan 0^\circ = 0\)
\(\tan 30^\circ = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}\)
\(\tan 45^\circ = 1\)
\(\tan 60^\circ = \sqrt{3}\)
\(\tan 90^\circ\) is undefined
Section 2 Key Takeaway: The CAST diagram gives you the sign of each ratio in any quadrant, while exact values allow you to write algebraic solutions without rounding errors.
3. Fundamental Trigonometric Identities
Identities are mathematical statements that are true for every value of \(\theta\). In Unit AS 1, two fundamental identities are examined:
Identity 1: The Quotient Identity
\(\tan \theta \equiv \frac{\sin \theta}{\cos \theta} \quad (\text{for } \cos \theta \neq 0)\)
Identity 2: The Pythagorean Identity
\(\sin^2 \theta + \cos^2 \theta \equiv 1\)
This identity can be rearranged in two useful ways:
• \(\sin^2 \theta \equiv 1 - \cos^2 \theta\)
• \(\cos^2 \theta \equiv 1 - \sin^2 \theta\)
Worked Example: Simplifying Expressions
Problem: Simplify the expression \(\frac{1 - \cos^2 \theta}{\sin \theta \cos \theta}\).
Step 1: Replace the numerator using the Pythagorean identity: \(1 - \cos^2 \theta = \sin^2 \theta\).
Step 2: Substitute back into the fraction: \(\frac{\sin^2 \theta}{\sin \theta \cos \theta}\).
Step 3: Cancel a common factor of \(\sin \theta\): \(\frac{\sin \theta}{\cos \theta}\).
Step 4: Apply the quotient identity: \(\frac{\sin \theta}{\cos \theta} = \tan \theta\).
Section 3 Key Takeaway: Look for \(\sin^2\theta\) or \(\cos^2\theta\) to apply \(\sin^2\theta + \cos^2\theta \equiv 1\), and look for \(\frac{\sin\theta}{\cos\theta}\) to convert into \(\tan\theta\).
4. Solving Trigonometric Equations
Trigonometric equations typically have multiple solutions within a specified interval (e.g., \(0^\circ \le \theta \le 360^\circ\) or \(-180^\circ \le \theta \le 180^\circ\)). Follow these standard methods depending on the structure of the equation.
Type A: Linear Equations with Transformed Angles
To solve equations of the form \(\sin(k\theta + \alpha) = c\), \(\cos(k\theta + \alpha) = c\), or \(\tan(k\theta + \alpha) = c\):
1. Transform the interval: If the range is \(0^\circ \le \theta \le 360^\circ\), adjust the range for \(u = k\theta + \alpha\). For example, for \(2\theta\), the interval becomes \(0^\circ \le 2\theta \le 720^\circ\).
2. Find the principal value: Calculate \(u_1 = \sin^{-1}(c)\) using your calculator.
3. Find all values of \(u\) within the transformed interval using CAST or symmetry rules.
4. Solve for \(\theta\): Undo the transformation (subtract \(\alpha\) and divide by \(k\)).
Type B: Equations Equating Sine and Cosine
When an equation has the form \(a\sin\theta = b\cos\theta\):
1. Divide both sides by \(\cos \theta\) (where \(\cos \theta \neq 0\)):
\(\frac{a\sin\theta}{\cos\theta} = b \implies a\tan\theta = b\)
2. Rearrange to get \(\tan\theta = \frac{b}{a}\).
3. Solve for \(\theta\) within the given interval.
Type C: Quadratic Trigonometric Equations
Equations with a mixture of squared terms and single terms (e.g., \(2\cos^2\theta - \sin\theta - 1 = 0\)) are solved by turning them into quadratics in a single trigonometric function:
1. Substitute \(\cos^2\theta = 1 - \sin^2\theta\) (or \(\sin^2\theta = 1 - \cos^2\theta\)).
2. Expand and collect terms to form a quadratic equation: \(a\sin^2\theta + b\sin\theta + c = 0\).
3. Factorise or use the quadratic formula to solve for \(\sin\theta\).
4. Solve each resulting linear equation within the required interval (discard any values where \(|\sin\theta| > 1\) or \(|\cos\theta| > 1\)).
Section 4 Key Takeaway: Always adjust the interval for transformed angles before finding roots, and convert quadratics into a single trig ratio using \(\sin^2\theta + \cos^2\theta \equiv 1\).
5. Top Examiner Pitfalls & Revision Checklist
Keep these frequent CCEA exam pitfalls in mind to secure top marks:
• Never divide through by a trigonometric function: In an equation such as \(\sin\theta\cos\theta = 2\sin\theta\), dividing both sides by \(\sin\theta\) loses the valid solutions where \(\sin\theta = 0\). Instead, rearrange to \(\sin\theta\cos\theta - 2\sin\theta = 0\) and factorise: \(\sin\theta(\cos\theta - 2) = 0\).
• Missing Solutions from Scaled Angles: When solving \(\cos(2\theta) = 0.5\) for \(0^\circ \le \theta \le 360^\circ\), there are 4 solutions across \(0^\circ \le 2\theta \le 720^\circ\). Finding only 2 solutions is a common error.
• Premature Rounding: Keep unrounded values in your calculator memory during multi-step calculations. Rounding intermediate values leads to inaccurate final answers.
• Degree Mode: Ensure your calculator is set to Degrees (D) mode for all AS 1 examination questions.
• Accuracy Conventions: Unless specified otherwise, give non-exact angle answers to 3 significant figures (or 1 decimal place if requested in degrees).