Historical Context & Motivation
Trigonometry has been a tool for understanding angles and distances for thousands of years, but the idea of relating the sine or cosine of a double angle (like 2θ) back to the original angle θ took centuries to develop. Ancient astronomers needed these relationships to predict planetary positions and eclipses, and over time mathematicians formalized the identities we use today. These double angle identities are not just algebraic tricks — they simplify complex calculations and reveal hidden patterns in trigonometric graphs.
The central question this lesson addresses is: how can we express sin 2θ and cos 2θ in terms of sin θ and cos θ, and what does doubling the angle do to a trigonometric graph? By the end of this lesson, you will be able to apply the double angle identities algebraically and predict how they transform the shape, period, and amplitude of sine and cosine curves.
Core Principles & Definitions
Before diving into formulas, let's establish the foundational ideas. The double angle identities are derived from the compound angle formulas (also called addition formulas) by setting both angles equal. In other words, sin(A + B) becomes sin(θ + θ) = sin 2θ. This simple substitution unlocks three powerful relationships that connect expressions involving 2θ to expressions involving just θ.
Double Angle for Sine
Double Angle for Cosine
Three Forms of cos 2θ
Effect on Graphs
Visual Explanation — Sine and Sin 2θ Compared
Look carefully at the diagram above. The violet sine curve completes one full oscillation from 0 to 2π, which is the standard period of sin θ. Now compare it to the cyan sin 2θ curve, which completes two full oscillations over the same interval. The amplitude (the height from the center line to the peak) stays at 1 for both curves. The only change is in the period: doubling the angle inside the sine function cuts the period in half. In general, the period of sin(Bθ) is 2π ÷ B, so for sin 2θ the period is 2π ÷ 2 = π.
Mathematical Framework
The double angle identities are derived from the addition formulas for sine and cosine. Start with sin(A + B) = sin A cos B + cos A sin B and set A = B = θ. Doing the same for cosine and applying the Pythagorean identity sin²θ + cos²θ = 1 yields three equivalent forms for cos 2θ.
Graph Transformations with Double Angles
Understanding how the double angle changes a graph is just as important as the algebra. When you replace θ with 2θ inside a trig function, you perform a horizontal compression by a factor of 2. Let's compare the key features of y = cos θ and y = cos 2θ side by side to make this concrete.
| Feature | y = cos θ | y = cos 2θ |
|---|---|---|
| Amplitude | 1 | 1 |
| Period | 2π ≈ 6.28 | π ≈ 3.14 |
| Zeros in [0, 2π] | π/2, 3π/2 | π/4, 3π/4, 5π/4, 7π/4 |
| Maximum points in [0, 2π] | θ = 0, 2π | θ = 0, π, 2π |
| Minimum points in [0, 2π] | θ = π | θ = π/2, 3π/2 |
Notice a pattern: the number of zeros, maxima, and minima all double when we go from cos θ to cos 2θ within the interval [0, 2π]. This makes sense because the period is halved, so the curve fits two complete cycles into the same window. The general rule is that for y = sin(Bθ) or y = cos(Bθ), the period equals 2π/B.
Worked Example
Let's work through a full problem that combines the double angle identities with algebraic reasoning, the kind of question you'll see on an IB exam.
Choosing the Right Form of cos 2θ
One of the trickiest parts of double angle problems is deciding which of the three forms of cos 2θ to use. The choice depends entirely on what information you're given or what you're trying to achieve in a simplification. Here's a guide to help you make the right pick every time.
| Form of cos 2θ | Best Used When… | Example Scenario |
|---|---|---|
| cos²θ − sin²θ | You know both sin θ and cos θ | Given sin θ = 3/5 in Q1, you find cos θ = 4/5, then use both. |
| 2cos²θ − 1 | You only know cos θ, or the expression involves only cos terms | Simplify an equation that already has cos²θ terms. |
| 1 − 2sin²θ | You only know sin θ, or the expression involves only sin terms | Given sin θ = 0.6, directly compute cos 2θ without finding cos θ. |
Connection to Advanced Topics
The double angle identities you've learned here are a gateway to more advanced trigonometric work. In the IB HL course and beyond, you will encounter half-angle formulas, power-reduction identities, and even triple-angle formulas — all of which build directly on the double angle identities. Additionally, the idea of analyzing how coefficients inside trig functions affect graphs extends to the general sinusoidal model y = a sin(b(x − c)) + d.
| What You Know Now (SL 3.6) | What Comes Next |
|---|---|
| sin 2θ = 2 sin θ cos θ | Generalise to sin 3θ, or use in integration (HL Calculus) |
| cos 2θ = 1 − 2sin²θ | Rearrange to get sin²θ = (1 − cos 2θ)/2 (power-reduction) |
| Period of sin 2θ is π | Model real phenomena like sound beats, alternating current, tides |
| Choosing the right form of cos 2θ | Solving trig equations analytically (SL 3.8 and HL) |
In physics, the double angle identities appear when analyzing projectile motion (the range formula R = v²sin 2θ / g), wave interference, and oscillating circuits. Mastering these identities now will pay dividends across multiple subjects.
Practice Problems
Lesson Summary
The double angle identities express trig functions of 2θ in terms of θ: sin 2θ = 2 sin θ cos θ and cos 2θ = cos²θ − sin²θ, with two alternative forms 2cos²θ − 1 and 1 − 2sin²θ. These are derived from the compound angle (addition) formulas by setting both angles equal to θ, and they are provided in the IB formula booklet.
On a graph, replacing θ with 2θ inside sin or cos produces a horizontal compression that halves the period from 2π to π while leaving the amplitude unchanged. When solving problems, choose the form of cos 2θ that matches the information you're given — use 2cos²θ − 1 when you know cos θ, 1 − 2sin²θ when you know sin θ, and always verify your answers using sin²(2θ) + cos²(2θ) = 1.