Trig identities – Pythagorean, compound angle, double angle and R cos(θ − α)

What it shows

Each identity is built from right-angled triangles. On the unit circle the triangles with sides cos θ, sin θ, tan θ and cot θ give the three Pythagorean identities. Two right triangles stacked inside a rectangle, with angle α and then β and a hypotenuse of 1, give sin(α ± β) and cos(α ± β) as the coordinates of P; dividing them gives tan(α ± β). A triangle in a semicircle gives the double-angle formulae. The wave screen shows that a cos θ + b sin θ is a single sinusoid of amplitude R = √(a² + b²). The diagrams use angles from 0° to 90°, but the formulae hold for all angles.

How to use

Choose a screen. On Pythagorean, drag P or the θ slider and press an identity to highlight its triangle. On Compound angles, drag Q to change α and P to change β, and switch between α + β and α − β. On a cos θ + b sin θ, set a and b and drag along the graph. On Identity checker, choose an Example or type the LHS and RHS.

Parameters you can change

• Screen Pythagorean, Compound angles, Double angle, a cos θ + b sin θ, Identity checker
• Angle θ (Pythagorean, Double angle, a cos θ + b sin θ screens) 0–360 °
• Angle α (Compound angles screen) 0–90 °
• Angle β (Compound angles screen) 0–90 °
• Operation (Compound angles screen) α + β, α − β
• Coefficient a of cos θ -5–5
• Coefficient b of sin θ -5–5
• Harmonic form R cos(θ − α), R sin(θ + α)
• Example identity (Identity checker screen) sin²x + cos²x = 1, 1 + tan²x = sec²x, 1 + cot²x = cosec²x, sin 2x = 2 sin x cos x, cos 2x = 1 − 2sin²x, tan 2x = 2tan x / (1 − tan²x), sin(x + π/3) = sin x cos π/3 + cos x sin π/3, cos(x − π/4) = cos x cos π/4 + sin x sin π/4, 3cos x + 4sin x = 5cos(x − α), tan α = 4/3, tan x + cot x = sec x cosec x, (1 − cos 2x)/sin 2x = tan x, sin 2x = 2sin x ?, cos(x + π/4) = cos x + cos π/4 ?, (sin x + cos x)² = 1 ?

Questions to explore

1. In the rectangle, which segments equal sin α cos β and cos α sin β, and why do they add up to sin(α + β)?
2. For a = 3 and b = 4, what is the maximum value of 3 cos θ + 4 sin θ, and at which angle does it occur?
3. Why is cos(x + π/4) = cos x + cos π/4 not an identity, and what counterexample can you find?