T-4 · Trigonometry

The circle Pythagoras draws.

A point on the unit circle is a right triangle with hypotenuse 1, so sin²θ + cos²θ = 1 is Pythagoras. The exact values are not looked up. Two triangles give 30° and 45°, and every multiple of 15° follows by rotation, in exact arithmetic. Turn the circle into a wave, then leave the circle entirely for any triangle at all.

θ = 45° = π/4

P = (cos θ, sin θ) lies on the circle x² + y² = 1, quadrant I.

sin θ	√2/2 ≈ 0.707107
cos θ	√2/2 ≈ 0.707107
tan θ	1 ≈ 1

Identities, checked

θ is a multiple of 15°, so each identity is evaluated in ℚ(√2, √3): a zero is a true zero.

sin²θ + cos²θ = 1	residual … Pythagoras on the triangle with hypotenuse 1
1 + tan²θ = sec²θ	residual …
1 + cot²θ = csc²θ	residual …
sin 2θ = 2 sin θ cos θ	residual …
cos 2θ = cos²θ − sin²θ	residual …

Where the exact values come from

sin 30° = 1/2 is half an equilateral triangle, and cos 30° = √(1 − 1/4) = √3/2 by Pythagoras. The isosceles right triangle gives sin 45° = √(1/2) = √2/2. Then 15° = 45° − 30°, and every multiple of 15° follows by rotating 15° at a time. Nothing below is looked up.

sin

cos

tan

0°

15°

(√6 − √2)/4

(√2 + √6)/4

2 − √3

30°

1/2

√3/2

√3/3

45°

√2/2

60°

√3/2

1/2

√3

75°

(√2 + √6)/4

(√6 − √2)/4

2 + √3

90°

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Reference angle 45°. In quadrant I the signs are sin +, cos +, tan +.