Exact trigonometric values

In mathematics, the values of the trigonometric functions can be expressed approximately, as in ${\displaystyle \cos(\pi /4)\approx 0.707}$, or exactly, as in ${\displaystyle \cos(\pi /4)={\sqrt {2}}/2}$. While trigonometric tables contain many approximate values, the exact values for certain angles can be expressed by a combination of arithmetic operations and square roots. The angles with trigonometric values that are expressible in this way are exactly those that can be constructed with a compass and straight edge, and the values are called constructible numbers.

Common angles

The trigonometric functions of angles that are multiples of 15°, 18°, or 22.5° have simple algebraic values. These values are listed in the following table for angles from 0° to 45°.[1] In the table below, the label "Undefined" represents a ratio ${\displaystyle 1:0.}$ If the codomain of the trigonometric functions is taken to be the real numbers these entries are undefined, whereas if the codomain is taken to be the projectively extended real numbers, these entries take the value ${\displaystyle \infty }$ (see division by zero).

Radians Degrees sin cos tan cot sec csc
${\displaystyle 0}$ ${\displaystyle 0^{\circ }}$ ${\displaystyle 0}$ ${\displaystyle 1}$ ${\displaystyle 0}$ Undefined ${\displaystyle 1}$ Undefined
${\displaystyle {\frac {\pi }{12}}}$ ${\displaystyle 15^{\circ }}$ ${\displaystyle {\frac {\sqrt {2-{\sqrt {3}}}}{2}}}$ ${\displaystyle {\frac {\sqrt {2+{\sqrt {3}}}}{2}}}$ ${\displaystyle 2-{\sqrt {3}}}$ ${\displaystyle 2+{\sqrt {3}}}$ ${\displaystyle {\sqrt {2}}({\sqrt {3}}-1)}$ ${\displaystyle {\sqrt {2}}({\sqrt {3}}+1)}$
${\displaystyle {\frac {\pi }{10}}}$ ${\displaystyle 18^{\circ }}$ ${\displaystyle {\frac {{\sqrt {5}}-1}{4}}}$ ${\displaystyle {\frac {{\sqrt {2}}{\sqrt {5+{\sqrt {5}}}}}{4}}}$ ${\displaystyle {\frac {{\sqrt {5}}{\sqrt {5-2{\sqrt {5}}}}}{5}}}$ ${\displaystyle {\sqrt {5+2{\sqrt {5}}}}}$ ${\displaystyle {\frac {{\sqrt {10}}{\sqrt {5-{\sqrt {5}}}}}{5}}}$ ${\displaystyle {\sqrt {5}}+1}$
${\displaystyle {\frac {\pi }{8}}}$ ${\displaystyle 22.5^{\circ }}$ ${\displaystyle {\frac {\sqrt {2-{\sqrt {2}}}}{2}}}$ ${\displaystyle {\frac {\sqrt {2+{\sqrt {2}}}}{2}}}$ ${\displaystyle {\sqrt {2}}-1}$ ${\displaystyle {\sqrt {2}}+1}$ ${\displaystyle {\sqrt {2}}{\sqrt {2-{\sqrt {2}}}}}$ ${\displaystyle {\sqrt {2}}{\sqrt {2+{\sqrt {2}}}}}$
${\displaystyle {\frac {\pi }{6}}}$ ${\displaystyle 30^{\circ }}$ ${\displaystyle {\frac {1}{2}}}$ ${\displaystyle {\frac {\sqrt {3}}{2}}}$ ${\displaystyle {\frac {\sqrt {3}}{3}}}$ ${\displaystyle {\sqrt {3}}}$ ${\displaystyle {\frac {2{\sqrt {3}}}{3}}}$ ${\displaystyle 2}$
${\displaystyle {\frac {\pi }{5}}}$ ${\displaystyle 36^{\circ }}$ ${\displaystyle {\frac {{\sqrt {2}}{\sqrt {5-{\sqrt {5}}}}}{4}}}$ ${\displaystyle {\frac {{\sqrt {5}}+1}{4}}}$ ${\displaystyle {\sqrt {5-2{\sqrt {5}}}}}$ ${\displaystyle {\frac {{\sqrt {5}}{\sqrt {5+2{\sqrt {5}}}}}{5}}}$ ${\displaystyle {\sqrt {5}}-1}$ ${\displaystyle {\frac {{\sqrt {10}}{\sqrt {5+{\sqrt {5}}}}}{5}}}$
${\displaystyle {\frac {\pi }{4}}}$ ${\displaystyle 45^{\circ }}$ ${\displaystyle {\frac {\sqrt {2}}{2}}}$ ${\displaystyle 1}$ ${\displaystyle {\sqrt {2}}}$

For angles outside of this range, trigonometric values can be found by applying reflection and shift identities such as

{\displaystyle {\begin{alignedat}{3}&&\sin({\tfrac {\pi }{2}}-\theta )&{}=\cos(\theta ),\\[5mu]&&\sin(2\pi +\theta )&{}=\sin(\pi -\theta )&&{}=\sin(\theta ),\quad &&\sin(\pi +\theta )&&{}=\sin(-\theta )&&{}=-\sin(\theta ),\\[5mu]&&\cos(2\pi +\theta )&{}=\cos(-\theta )&&{}=\cos(\theta ),\quad &&\cos(\pi +\theta )&&{}=\cos(\pi -\theta )&&{}=-\cos(\theta ).\end{alignedat}}}

Trigonometric numbers

A trigonometric number is a number that can be expressed as the sine or cosine of a rational multiple of π radians.[2] Since ${\displaystyle \sin(x)=\cos(x-\pi /2),}$ the case of a sine can be omitted from this definition. Therefore any trigonometric number can be written as ${\displaystyle \cos(2\pi k/n)}$, where k and n are integers. This number can be thought of as the real part of the complex number ${\displaystyle \cos(2\pi k/n)+i\sin(2\pi k/n)}$. De Moivre's formula shows that numbers of this form are roots of unity:

${\displaystyle \left(\cos \left({\frac {2\pi k}{n}}\right)+i\sin \left({\frac {2\pi k}{n}}\right)\right)^{n}=\cos(2\pi k)+i\sin(2\pi k)=1}$

Since the root of unity is a root of the polynomial xn − 1, it is algebraic. Since the trigonometric number is the average of the root of unity and its complex conjugate, and algebraic numbers are closed under arithmetic operations, every trigonometric number is algebraic.[2] The minimal polynomials of trigonometric numbers can be explicitly enumerated.[3] In contrast, by the Lindemann–Weierstrass theorem, the sine or cosine of any non-zero algebraic number is always transcendental.[4]

The real part of any root of unity is a trigonometric number. By Niven's theorem, the only rational trigonometric numbers are 0, 1, −1, 1/2, and −1/2.[5]

Constructibility

An angle can be constructed with a compass and straightedge if and only if its sine (or equivalently cosine) can be expressed by a combination of arithmetic operations and square roots applied to integers.[6] Additionally, an angle that is a rational multiple of ${\displaystyle \pi }$ radians is constructible if and only if, when it is expressed as ${\displaystyle a\pi /b}$ radians, where a and b are relatively prime integers, the prime factorization of the denominator, b, is the product of some power of two and any number of distinct Fermat primes (a Fermat prime is a prime number one greater than a power of two).[7]

Thus, for example, ${\displaystyle 2\pi /15=24^{\circ }}$ is a constructible angle because 15 is the product of the Fermat primes 3 and 5. Similarly ${\displaystyle \pi /12=15^{\circ }}$ is a constructible angle because 12 is a power of two (4) times a Fermat prime (3). But ${\displaystyle \pi /9=20^{\circ }}$ is not a constructible angle, since ${\displaystyle 9=3\cdot 3}$ is not the product of distinct Fermat primes as it contains 3 as a factor twice, and neither is ${\displaystyle \pi /7\approx 25.714^{\circ }}$, since 7 is not a Fermat prime.[8]

It results from the above characterisation that an angle of an integer number of degrees is constructible if and only if this number of degrees is a multiple of 3.

Constructible values

45°

From a reflection identity, ${\displaystyle \cos(45^{\circ })=\sin(90^{\circ }-45^{\circ })=\sin(45^{\circ })}$. Substituting into the Pythagorean trigonometric identity ${\displaystyle \sin(45^{\circ })^{2}+\cos(45^{\circ })^{2}=1}$, one obtains the minimal polynomial ${\displaystyle 2\sin(45^{\circ })^{2}-1=0}$. Taking the positive root, one finds ${\displaystyle \sin(45^{\circ })=\cos(45^{\circ })=1/{\sqrt {2}}={\sqrt {2}}/2}$.

30° and 60°

The values of sine and cosine of 30 and 60 degrees are derived by analysis of the equilateral triangle. In an equilateral triangle, the 3 angles are equal and sum to 180°, therefore each corner angle is 60°. Bisecting one corner, the special right triangle with angles 30-60-90 is obtained. By symmetry, the bisected side is half of the side of the equilateral triangle, so one concludes ${\displaystyle \sin(30^{\circ })=1/2}$. The Pythagorean and reflection identities then give ${\displaystyle \sin(60^{\circ })=\cos(30^{\circ })={\sqrt {1-(1/2)^{2}}}={\sqrt {3}}/2}$.

18°, 36°, 54°, and 72°

The value of ${\displaystyle \sin(18^{\circ })}$ may be derived using the multiple angle formulas for sine and cosine.[9] By the double angle formula for sine:

${\displaystyle \sin(36^{\circ })=2\sin(18^{\circ })\cos(18^{\circ })}$

By the triple angle formula for cosine:

${\displaystyle \cos(54^{\circ })=\cos ^{3}(18^{\circ })-3\sin ^{2}(18^{\circ })\cos(18^{\circ })=\cos(18^{\circ })(1-4\sin ^{2}(18^{\circ }))}$

Since sin(36°) = cos(54°), we equate these two expressions and cancel a factor of cos(18°):

${\displaystyle 2\sin(18^{\circ })=1-4\sin ^{2}(18^{\circ })}$

This quadratic equation has only one positive root:

${\displaystyle \sin(18^{\circ })={\frac {{\sqrt {5}}-1}{4}}}$

The Pythagorean identity then gives ${\displaystyle \cos(18^{\circ })}$, and the double and triple angle formulas give sine and cosine of 36°, 54°, and 72°.

Remaining multiples of 3°

Wikimedia Commons has a file available for a table of these exact values.

The sines and cosines of all other angles between 0 and 90° that are multiples of 3° can be derived from the angles described above and the sum and difference formulas. Specifically,[10]

{\displaystyle {\begin{aligned}3^{\circ }&=18^{\circ }-15^{\circ },&24^{\circ }&=54^{\circ }-30^{\circ },&51^{\circ }&=60^{\circ }-9^{\circ },&78^{\circ }&=60^{\circ }+18^{\circ },&\\6^{\circ }&=36^{\circ }-30^{\circ },&27^{\circ }&=45^{\circ }-18^{\circ },&57^{\circ }&=30^{\circ }+27^{\circ },&81^{\circ }&=45^{\circ }+36^{\circ },&\\9^{\circ }&=45^{\circ }-36^{\circ },&33^{\circ }&=60^{\circ }-27^{\circ },&63^{\circ }&=45^{\circ }+18^{\circ },&84^{\circ }&=54^{\circ }+30^{\circ },&\\12^{\circ }&=30^{\circ }-18^{\circ },&39^{\circ }&=30^{\circ }+9^{\circ },&66^{\circ }&=36^{\circ }+30^{\circ },&87^{\circ }&=60^{\circ }+27^{\circ }.&\\15^{\circ }&=45^{\circ }-30^{\circ },&42^{\circ }&=60^{\circ }-18^{\circ },&69^{\circ }&=60^{\circ }+9^{\circ },&\\21^{\circ }&=30^{\circ }-9^{\circ },&48^{\circ }&=30^{\circ }+18^{\circ },&75^{\circ }&=45^{\circ }+30^{\circ },&\end{aligned}}}

For example, since ${\displaystyle 24^{\circ }=60^{\circ }-36^{\circ }}$, its cosine can be derived by the cosine difference formula:

{\displaystyle {\begin{aligned}\cos(24^{\circ })&=\cos(60^{\circ })\cos(36^{\circ })+\sin(60^{\circ })\sin(36^{\circ })\\&={\frac {1}{2}}{\frac {{\sqrt {5}}+1}{4}}+{\frac {\sqrt {3}}{2}}{\frac {\sqrt {10-2{\sqrt {5}}}}{4}}\\&={\frac {1+{\sqrt {5}}+{\sqrt {30-6{\sqrt {5}}}}}{8}}\end{aligned}}}

Half angles

If the denominator, b, is multiplied by additional factors of 2, the sine and cosine can be derived with the half-angle formulas. For example, 22.5° (π/8 rad) is half of 45°, so its sine and cosine are:[11]

${\displaystyle \sin(22.5^{\circ })={\sqrt {\frac {1-\cos(45^{\circ })}{2}}}={\sqrt {\frac {1-{\frac {\sqrt {2}}{2}}}{2}}}={\frac {1}{2}}{\sqrt {2-{\sqrt {2}}}}}$
${\displaystyle \cos(22.5^{\circ })={\sqrt {\frac {1+\cos(45^{\circ })}{2}}}={\sqrt {\frac {1+{\frac {\sqrt {2}}{2}}}{2}}}={\frac {1}{2}}{\sqrt {2+{\sqrt {2}}}}}$

Repeated application of the half-angle formulas leads to nested radicals, specifically nested square roots of 2 of the form ${\displaystyle {\sqrt {2\pm \cdots }}}$. In general, the sine and cosine of most angles of the form ${\displaystyle \beta /2^{n}}$ can be expressed using nested square roots of 2 in terms of ${\displaystyle \beta }$. Specifically, if one can write an angle as ${\displaystyle \alpha =\pi \left({\frac {1}{2}}-\sum _{i=1}^{k}{\frac {\prod _{j=1}^{i}b_{j}}{2^{i+1}}}\right)=\pi \left({\frac {1}{2}}-{\frac {b_{1}}{4}}-{\frac {b_{1}b_{2}}{8}}-{\frac {b_{1}b_{2}b_{3}}{16}}-\ldots -{\frac {b_{1}b_{2}\ldots b_{k}}{2^{k+1}}}\right)}$ where ${\displaystyle b_{k}\in [-2,2]}$ and ${\displaystyle b_{i}}$ is -1, 0, or 1 for ${\displaystyle i, then[12] ${\displaystyle \cos(\alpha )={\frac {b_{1}}{2}}{\sqrt {2+b_{2}{\sqrt {2+b_{3}{\sqrt {2+\ldots +b_{k-1}{\sqrt {2+2\sin \left({\frac {\pi b_{k}}{4}}\right)}}}}}}}}}$ and if ${\displaystyle b_{1}\neq 0}$ then[12] ${\displaystyle \sin(\alpha )={\frac {1}{2}}{\sqrt {2-b_{2}{\sqrt {2+b_{3}{\sqrt {2+b_{4}{\sqrt {2+\ldots +b_{k-1}{\sqrt {2+2\sin \left({\frac {\pi b_{k}}{4}}\right)}}}}}}}}}}}$ For example, ${\displaystyle {\frac {13\pi }{32}}=\pi \left({\frac {1}{2}}-{\frac {1}{4}}+{\frac {1}{8}}+{\frac {1}{16}}-{\frac {1}{32}}\right)}$, so one has ${\displaystyle (b_{1},b_{2},b_{3},b_{4})=(1,-1,1,-1)}$ and obtains: ${\displaystyle \cos \left({\frac {13\pi }{32}}\right)={\frac {1}{2}}{\sqrt {2-{\sqrt {2+{\sqrt {2+2\sin \left({\frac {-\pi }{4}}\right)}}}}}}={\frac {1}{2}}{\sqrt {2-{\sqrt {2+{\sqrt {2-{\sqrt {2}}}}}}}}}$ ${\displaystyle \sin \left({\frac {13\pi }{32}}\right)={\frac {1}{2}}{\sqrt {2+{\sqrt {2+{\sqrt {2-{\sqrt {2}}}}}}}}}$

Denominator of 17

Since 17 is a Fermat prime, a regular 17-gon is constructible, which means that the sines and cosines of angles such as ${\displaystyle 2\pi /17}$ radians can be expressed in terms of square roots. In particular, in 1796, Carl Friedrich Gauss showed that:[13][14]

${\displaystyle \cos \left({\frac {2\pi }{17}}\right)={\frac {-1+{\sqrt {17}}+{\sqrt {34-2{\sqrt {17}}}}+2{\sqrt {17+3{\sqrt {17}}-{\sqrt {170+38{\sqrt {17}}}}}}}{16}}}$

The sines and cosines of other constructible angles of the form ${\displaystyle {\frac {k2^{n}\pi }{17}}}$ (for integers ${\displaystyle k,n}$) can be derived from this one.

Non-constructibility of 1°

As discussed in § Constructibility, only certain angles that are rational multiples of ${\displaystyle \pi }$ radians have trigonometric values that can be expressed with square roots. The angle 1°, being ${\displaystyle \pi /180=\pi /(2^{2}\cdot 3^{2}\cdot 5)}$ radians, has a repeated factor of 3 in the denominator and therefore ${\displaystyle \sin(1^{\circ })}$ cannot be expressed using only square roots. A related question is whether it can be expressed using cube roots. The following two approaches can be used, but both result in an expression that involves the cube root of a complex number.

Using the triple-angle identity, we can identify ${\displaystyle \sin(1^{\circ })}$ as a root of a cubic polynomial: ${\displaystyle \sin(3^{\circ })=-4x^{3}+3x}$. The three roots of this polynomial are ${\displaystyle \sin(1^{\circ })}$, ${\displaystyle \sin(59^{\circ })}$, and ${\displaystyle -\sin(61^{\circ })}$. Since ${\displaystyle \sin(3^{\circ })}$ is constructible, an expression for it could be plugged into Cardano's formula to yield an expression for ${\displaystyle \sin(1^{\circ })}$. However, since all three roots of the cubic are real, this is an instance of casus irreducibilis, and the expression would require taking the cube root of a complex number.[15][16]

Alternatively, by De Moivre's formula:

{\displaystyle {\begin{aligned}(\cos(1^{\circ })+i\sin(1^{\circ }))^{3}&=\cos(3^{\circ })+i\sin(3^{\circ }),\\[4mu](\cos(1^{\circ })-i\sin(1^{\circ }))^{3}&=\cos(3^{\circ })-i\sin(3^{\circ }).\end{aligned}}}

Taking cube roots and adding or subtracting the equations, we have:[16]

{\displaystyle {\begin{aligned}\cos(1^{\circ })&=\;{\frac {1}{2}}\left({\sqrt[{3}]{\cos(3^{\circ })+i\sin(3^{\circ })}}+{\sqrt[{3}]{\cos(3^{\circ })-i\sin(3^{\circ })}}\right),\\[5mu]\sin(1^{\circ })&={\frac {1}{2i}}\left({\sqrt[{3}]{\cos(3^{\circ })+i\sin(3^{\circ })}}-{\sqrt[{3}]{\cos(3^{\circ })-i\sin(3^{\circ })}}\right).\end{aligned}}}

References

1. ^ Abramowitz & Stegun 1972, p. 74, 4.3.46
2. ^ a b Niven, Ivan. Numbers: Rational and Irrational, 1961. Random House. New Mathematical Library, Vol. 1. ISSN 0548-5932. Ch. 5
3. ^ Lehmer, D. H. (1933). "A Note on Trigonometric Algebraic Numbers". The American Mathematical Monthly. 40 (3): 165–166. doi:10.2307/2301023. JSTOR 2301023.
4. ^ Burger, Edward B.; Tubbs, Robert (17 April 2013). Making Transcendence Transparent: An intuitive approach to classical transcendental number theory. Springer Science & Business Media. p. 44. ISBN 978-1-4757-4114-8.
5. ^ Schaumberger, Norman (1974). "A Classroom Theorem on Trigonometric Irrationalities". Two-Year College Mathematics Journal. 5 (1): 73–76. doi:10.2307/3026991. JSTOR 3026991.
6. ^ Martin, George E. (1998), Geometric Constructions, Undergraduate Texts in Mathematics, Springer-Verlag, New York, doi:10.1007/978-1-4612-0629-3, ISBN 0-387-98276-0, MR 1483895
7. ^ Martin, George E. (1998), Geometric Constructions, Undergraduate Texts in Mathematics, Springer-Verlag, New York, p. 46, doi:10.1007/978-1-4612-0629-3, ISBN 0-387-98276-0, MR 1483895
8. ^ Fraleigh, John B. (1994), A First Course in Abstract Algebra (5th ed.), Addison Wesley, ISBN 978-0-201-53467-2, MR 0225619
9. ^ "Exact Value of sin 18°". math-only-math.
10. ^ Weiß, Adam (1851). Handbuch Der Trigonometrie (in German). J. L. Schmid. pp. 72–74.
11. ^ Durbha, Subramanyam (2012). "A Geometric Method of Finding the Trigonometric Ratios of 22 ½° and 75°". Mathematics in School. 41 (3): 22–23. JSTOR 23269221.
12. ^ a b Servi, L. D. (April 2003). "Nested Square Roots of 2". The American Mathematical Monthly. 110 (4): 326–330. doi:10.1080/00029890.2003.11919968.
13. ^ Arthur Jones, Sidney A. Morris, Kenneth R. Pearson, Abstract Algebra and Famous Impossibilities, Springer, 1991, ISBN 0387976612, p. 178.
14. ^ Callagy, James J. "The central angle of the regular 17-gon", Mathematical Gazette 67, December 1983, 290–292.
15. ^ Parent, James T. (June 2011). "Exact values for the sin of all integers" (PDF). Interactive Mathematics. Retrieved 5 February 2024.
16. ^ a b Kowalski, Travis (November 2016). "The Sine of a Single Degree" (PDF). The College Mathematics Journal. 47 (5): 322–332. doi:10.4169/college.math.j.47.5.322. S2CID 125810699.