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IB DP Mathematics: Applications and Interpretation | SL & HL | Geometry and Trigonometry — Topic 3.8
📋 Learning Intentions
- Define the unit circle and use it to determine the exact values of sine, cosine, and tangent for key angles in both degrees and radians.
- State and apply the Pythagorean identity \( \sin^2\theta + \cos^2\theta = 1 \) to find unknown trigonometric values given a constraint.
- Solve trigonometric equations of the form \( \sin\theta = k \), \( \cos\theta = k \), and \( \tan\theta = k \) over a specified interval, identifying all solutions.
- Interpret the symmetry properties of the unit circle to generate secondary solutions from a principal value.
- Use the GDC to verify solutions and graph trigonometric functions to identify intersections within a given domain.
- Justify the number of solutions to a trigonometric equation by reference to the period and symmetry of the relevant function.
Real-World Introduction
Engineers designing rotating machinery — from wind turbines to engine crankshafts — need to model how a point on a rotating component moves over time. The horizontal and vertical displacements of that point trace out cosine and sine curves respectively, completing one full cycle every revolution. Without a systematic way to find the exact angle at which a displacement reaches a target value, engineers cannot determine when a valve opens, when a piston reaches maximum compression, or when a turbine blade is at peak stress.
The unit circle provides the geometric foundation for all of this. By placing a circle of radius 1 at the origin, every angle \(\theta\) maps to a unique point \((\cos\theta,\, \sin\theta)\) on the circle. Solving a trigonometric equation is then equivalent to asking: at which angles does the rotating radius reach a given height or horizontal position? Because the circle is periodic and symmetric, there are always multiple answers within any full rotation — and the unit circle makes those symmetry patterns visible.
🌍 Why This Matters Tide prediction models, electrical AC circuit analysis, and sound wave engineering all require solving equations of the form \(\sin\theta = k\) or \(\cos\theta = k\) over extended intervals. The unit circle is the conceptual engine behind every such calculation.
Core Concepts & Definitions
Consider a circle centred at the origin with radius exactly 1. Any point \(P\) on this circle can be reached by rotating a radius from the positive \(x\)-axis through an angle \(\theta\). The coordinates of \(P\) depend entirely on \(\theta\), and this relationship defines the trigonometric functions for all angles — not just the acute angles of right-triangle trigonometry.
📐 Key Definition — The Unit Circle The unit circle is the circle with equation \( x^2 + y^2 = 1 \), centred at the origin with radius 1. For any angle \(\theta\) measured anticlockwise from the positive \(x\)-axis, the point \(P = (\cos\theta,\, \sin\theta)\) lies on the unit circle. The