Trigonometry: trig functions, identities, equations, solutions, graphs
Pure Mathematics 1: Trigonometry
1️⃣ Basic Trig Functions & the Unit Circle
Think of a clock face. The hour hand is like the radius of a circle that always stays the same length. As the hand turns, it traces a path – that’s the unit circle (radius = 1). The angle θ is measured from the positive x‑axis.
From any point on the unit circle:
- sin θ = y‑coordinate
- cos θ = x‑coordinate
- tan θ = sin θ / cos θ = y/x
📐 Example: For θ = 30°, sin θ = ½, cos θ = √3⁄2, tan θ = 1⁄√3.
2️⃣ Key Identities
These are the “rules of the road” for trig functions.
| Identity | What it means |
|---|---|
| $\\sin^2\\theta + \\cos^2\\theta = 1$ | Pythagorean identity – like the unit circle’s radius. |
| $\\tan\\theta = \\dfrac{\\sin\\theta}{\\cos\\theta}$ | Definition of tangent. |
| $\\sin(\\alpha\\pm\\beta) = \\sin\\alpha\\cos\\beta \\pm \\cos\\alpha\\sin\\beta$ | Angle addition/subtraction. |
| $\\cos(\\alpha\\pm\\beta) = \\cos\\alpha\\cos\\beta \\mp \\sin\\alpha\\sin\\beta$ | Angle addition/subtraction. |
| $\\sin 2\\theta = 2\\sin\\theta\\cos\\theta$ | Double‑angle for sine. |
| $\\cos 2\\theta = \\cos^2\\theta - \\sin^2\\theta$ | Double‑angle for cosine. |
🔍 Tip: Memorise the first two identities; the rest follow from them.
3️⃣ Solving Trig Equations
When you see an equation like $\\sin\\theta = \\dfrac{1}{2}$, you’re looking for all angles that satisfy it.
- Find the principal solution (the basic angle). For $\\sin\\theta = ½$, θ = 30° or θ = 150°.
- Use the periodicity of the function: $\\theta = \\theta_0 + 360°k$ for sine and cosine, $\\theta = \\theta_0 + 180°k$ for tangent.
- Combine both to get the general solution: $\\theta = 30° + 360°k$ or $\\theta = 150° + 360°k$.
💡 Example: Solve $\\cos\\theta = -\\dfrac{\\sqrt{3}}{2}$ in $0°\\le\\theta<360°$.
Answer: θ = 120° or θ = 240°.
🧠 Trick: Draw a quick unit‑circle sketch to visualise where the cosine value occurs.
4️⃣ Graphs of Trig Functions
A graph is like a roller‑coaster for your function.
- sin x – starts at 0, peaks at 90°, dips at 270°, repeats every 360°.
- cos x – starts at 1, dips at 180°, peaks at 360°, repeats every 360°.
- tan x – vertical asymptotes at 90° and 270°, repeats every 180°.
⚙️ Key features to note:
- Amplitude – the height from the midline (for sin and cos, amplitude = 1).
- Period – the horizontal length of one full cycle (360° for sin/cos, 180° for tan).
- Phase shift – horizontal shift; e.g., $\\sin(x-30°)$ starts 30° later.
- Vertical shift – moves the graph up/down; e.g., $\\cos x + 2$ lifts the whole wave by 2 units.
📌 Remember: The midline is the horizontal line the wave oscillates around.
📚 Exam Tips & Quick Checklist
- Always write the general solution when asked for “all solutions.”
- Check the domain specified in the question (e.g., $0°\\le\\theta<360°$).
- Use the unit circle to visualise angles and values.
- When simplifying, reduce fractions and rationalise denominators if required.
- Remember the periodic nature of trig functions to avoid missing solutions.
- For graph questions, label the amplitude, period, phase shift clearly.
- Practice converting between degrees and radians; the exam may mix both.
💬 Final thought: Trigonometry is like a musical instrument – the more you practice, the more natural the patterns become. 🎶
Revision
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