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Addition and half-angle identities compute the sine, cosine, and tangent of $π / 12$

Example

Using $π / 12 = π / 3 - π / 4$ , $\sin (π / 12) = \frac{\sqrt{6} - \sqrt{2}}{4}, \cos (π / 12) = \frac{\sqrt{6} + \sqrt{2}}{4}, \tan (π / 12) = 2 - \sqrt{3} .$

Facts & Assumptions

Given: The positive number $π$ and the angles $π / 4, π / 3 \in (0, π)$ .

[L1]

Quarter-turn values and shifts by pi/2 and pi gives $\sin (π / 2) = 1$ and $\cos (π / 2) = 0$ .

[L2]

Half-angle identities with the sign determined by the quadrant gives $\cos (x / 2) = ε_{c} \sqrt{(1 + \cos x) / 2}$ and $\sin (x / 2) = ε_{s} \sqrt{(1 - \cos x) / 2}$ , with the signs of the corresponding half-angle functions.

[L3]

Cofunction, supplementary, quarter-turn, and reflection identities for the six trigonometric functions gives $\cos (π / 2 - x) = \sin x$ and $\cos (π - x) = - \cos x$ .

[L4]

Double-angle and quadratic power-reduction identities gives $\cos (2 x) = 2 \cos^{2} x - 1$ .

[L5]

Pi is the first positive zero of sine gives $\sin x > 0$ whenever $0 < x < π$ .

[L6]

The subtraction formulas for sine and cosine gives $\sin (u - v) = \sin u \cos v - \cos u \sin v$ and $\cos (u - v) = \cos u \cos v + \sin u \sin v$ .

[L7]

Tangent, cotangent, secant, and cosecant on their exact natural domains defines $\tan x = \sin x / \cos x$ when $\cos x \neq 0$ .

Verification

1.1

By [L5], $\sin (π / 4) > 0$ , and [L3] gives $\cos (π / 4) = \sin (π / 4) > 0$ . Applying [L2] to $x = π / 2$ and using [L1] therefore gives $\sin (π / 4) = \cos (π / 4) = \sqrt{2} / 2$ .

L1L2L3L5algebra

1.2

Put $c = \cos (π / 3)$ . Then $c > 0$ because [L3] writes it as $\sin (π / 6)$ and [L5] makes that value positive. Also [L3] and [L4] give $- c = \cos (2 π / 3) = 2 c^{2} - 1$ , hence $(2 c - 1) (c + 1) = 0$ ; positivity rules out $c = - 1$ , so $\cos (π / 3) = 1 / 2$ .

L3L4L5algebra

2.1

From [L3] and step 1.2, $\cos (2 π / 3) = - 1 / 2$ . Since $\sin (π / 3) > 0$ by [L5], [L2] applied to $x = 2 π / 3$ gives $\sin (π / 3) = \sqrt{(1 + 1 / 2) / 2} = \sqrt{3} / 2$ .

L2L3L5step 1.2algebra

3.1

Substitute steps 1.1, 1.2, and 2.1 into [L6] with $u = π / 3$ and $v = π / 4$ . This gives $\sin (π / 12) = (\sqrt{6} - \sqrt{2}) / 4$ and $\cos (π / 12) = (\sqrt{6} + \sqrt{2}) / 4 > 0$ .

L6step 1.1step 1.2step 2.1algebra

4.1

By [L7] and step 3.1, $\tan (π / 12) = (\sqrt{6} - \sqrt{2}) / (\sqrt{6} + \sqrt{2})$ ; rationalizing gives $(8 - 4 \sqrt{3}) / 4 = 2 - \sqrt{3}$ .

L7step 3.1algebra∎

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Sources

NIST Digital Library of Mathematical Functions, Chapter 4 (standard reference, not scraped)

Addition and half-angle identities compute the sine, cosine, and tangent of π/12π/12