Exact trigonometric values

Content sourced from Wikipedia, licensed under CC BY-SA 3.0.

Exact trigonometric values explained simply

In trigonometry, some angles have exact values that can be written with ordinary numbers and square roots. These are called constructible values because such angles can be drawn with a compass and straightedge. Other angles only have approximate decimal values.

What makes an angle’s sine or cosine exact?
- If the angle is a rational multiple of pi (for example, 15°, 30°, 45°, etc.), sometimes its sine or cosine can be written using just arithmetic and square roots.
- An angle aπ/b is constructible (i.e., can be built with a compass and straightedge) if and only if b’s prime factors are 2 and any number of distinct Fermat primes (primes of the form 2^(2^k) + 1). This leads to many familiar angles having exact expressions.

Key examples of constructible angles (0° to 90°)
- 45°: sin 45° = cos 45° = √2/2 ≈ 0.7071
- 30° and 60°: sin 30° = 1/2, cos 30° = √3/2 ≈ 0.8660
- 15°: sin 15° = (√6 − √2)/4, cos 15° = (√6 + √2)/4
- 18°, 36°, 54°, 72°: these involve the golden ratio φ = (1 + √5)/2
- sin 18° = (√5 − 1)/4 ≈ 0.3090
- cos 36° = φ/2 = (√5 + 1)/4 ≈ 0.8090
- sin 36° = √(10 − 2√5)/4 ≈ 0.5880
- sin 54° = cos 36° ≈ 0.8090
- sin 72° = cos 18° = √(10 + 2√5)/4 ≈ 0.9511
- 22.5° (half of 45°): sin 22.5° = (1/2)√(2 − √2), cos 22.5° = (1/2)√(2 + √2)
- 24°: cos 24° can also be written exactly, for example cos 24° = (1 + √5 + √(30 − 6√5)) / 8

Other notable constructible examples
- 24° can be obtained as 60° − 36°, and its cosine uses a combination of square roots.
- The 17-gon is constructible: cos(2π/17) can be written using square roots (Gauss showed this), and related angles kπ/(17·2^n) are likewise expressible with square roots.

What about most other angles?
- Not all angles have simple square-root expressions. For example, 1° cannot be expressed using only square roots. In some approaches, expressing sin 1° leads to solving a cubic equation, which in general requires cube roots of complex numbers (a situation called casus irreducibilis).

Two useful ideas to build more values
- Use relationships like sin(x) = cos(π/2 − x) to switch between sine and cosine.
- Use sum, difference, and half-angle formulas to derive values of related angles (for example, deriving 24°, 36°, 72°, or 15° from combinations of 30°, 45°, and 60°).

In short
- Many special angles have exact expressions using only square roots, especially those whose denominators in aπ/b fit the constructibility rule.
- A few well-known results include sin and cos for 15°, 18°, 24°, 30°, 36°, 45°, 54°, 60°, and 72°.
- However, not every angle is constructible; some require more advanced (and not purely real) expressions, and some angles like 1° do not have a finite square-root-based expression.
- Overall, exact trigonometric values are abundant for special, constructible angles, but most angles must be handled with numerical approximations.