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Plane Trigonometry
 De Moivre's Theorem
 Expansion of cosπ‘›πœƒ, β‹― in powers sinπœƒ and cosπœƒ
 Expansion of sine and cosine in powers the angle
 Expansion of cosπ‘›πœƒ and sinπ‘›πœƒ in cosines or sines of multiples of πœƒ
 Expansion of cosπ‘›πœƒ and sinπ‘›πœƒ in powers of sinπœƒ
 Expansion of cosπ‘›πœƒ in descending powers of cosπœƒ
 Expansion of πœƒ in powers of tanπœƒ (Gregory's series)
 formula for the calculation of the value of πœ‹ by Gregor's series
 To Prove that πœ‹ is Incommensurable
 Expansion of the sine and cosine in factors
 De Moivre's Property of the Circle
 Cotes's Properties
 Sources and References

Plane Trigonometry

De Moivre's Theorem

756 (cos𝛼+𝑖sin𝛼)(cos𝛽+𝑖cos𝛽)β‹―=cos(𝛼+𝛽+𝛾+β‹―)+𝑖sin(𝛼+𝛽+𝛾+β‹―), where 𝑖=βˆ’1 Proved by Induction 757 (cosπœƒ+𝑖sinπœƒ)𝑛=cosπ‘›πœƒ+𝑖sinπ‘›πœƒ Proof: By Induction, or by putting 𝛼, 𝛽, β‹― each = πœƒ in (756).

Expansion of cosπ‘›πœƒ, β‹― in powers sinπœƒ and cosπœƒ

758 cosπ‘›πœƒ=cosπ‘›πœƒβˆ’πΆ(𝑛,2)cosπ‘›βˆ’2πœƒsin2πœƒ+𝐢(𝑛,4)cosπ‘›βˆ’4πœƒsin4πœƒβˆ’β‹― 759 sinπ‘›πœƒ=𝑛cosπ‘›βˆ’1πœƒsinπœƒβˆ’πΆ(𝑛,3)cosπ‘›βˆ’3πœƒsin3πœƒ+β‹― Proof: Expand (757) by Bin. Th., and equate real and imaginary parts. 760 tanπ‘›πœƒ=𝑛tanπœƒβˆ’πΆ(𝑛,3)tan3πœƒ+β‹―1βˆ’πΆ(𝑛,2)tan2πœƒ+𝐢(𝑛,4)tan4πœƒβˆ’β‹― In series (758, 759), stop at, and exclude, all terms with indices greater than 𝑛. Note, 𝑛 is here an integer. 761 Let π‘ π‘Ÿ=sum of the 𝐢(𝑛,π‘Ÿ) products of tan𝛼, tan𝛽, tan𝛾, β‹― to 𝑛 terms. sin(𝛼+𝛽+𝛾+β‹―)=cos𝛼cos𝛽⋯(𝑠1βˆ’π‘ 3+𝑠5βˆ’β‹―) 762 cos(𝛼+𝛽+𝛾+β‹―)=cos𝛼cos𝛽⋯(1βˆ’π‘ 2+𝑠4βˆ’β‹―) Proof: By equating real and imaginary parts in (756). 763 tan(𝛼+𝛽+𝛾+β‹―)=(𝑠1βˆ’π‘ 3+𝑠5βˆ’π‘ 7+β‹―)1βˆ’π‘ 2+𝑠4βˆ’π‘ 6+β‹―

Expansion of sine and cosine in powers the angle

764 sinπœƒ=πœƒβˆ’πœƒ33!+πœƒ55!βˆ’β‹―, cosπœƒ=1βˆ’πœƒ22!+πœƒ44!βˆ’β‹― Proof: Put πœƒπ‘› for πœƒ in (757) and 𝑛=∞, employing (754) and (755). 766 π‘’π‘–πœƒ=cosπœƒ+𝑖sinπœƒ, π‘’βˆ’π‘–πœƒ=cosπœƒβˆ’π‘–sinπœƒBy 150 768 π‘’π‘–πœƒ+π‘’βˆ’π‘–πœƒ=2cosπœƒ, π‘’π‘–πœƒβˆ’π‘’βˆ’π‘–πœƒ=2𝑖sinπœƒ 770 𝑖tanπœƒ=π‘’π‘–πœƒβˆ’π‘’βˆ’π‘–πœƒπ‘’π‘–πœƒ+π‘’βˆ’π‘–πœƒ, 1+𝑖tanπœƒ1βˆ’π‘–tanπœƒ=𝑒2π‘–πœƒ

Expansion of cosπ‘›πœƒ and sinπ‘›πœƒ in cosines or sines of multiples of πœƒ

772 2π‘›βˆ’1cosπ‘›πœƒ=cosπ‘›πœƒ+𝑛cos(π‘›βˆ’2)πœƒ+𝐢(𝑛,2)cos(π‘›βˆ’1)πœƒ+𝐢(𝑛,3)cos(π‘›βˆ’6)πœƒ+β‹― 773 When 𝑛 is even, 2π‘›βˆ’1(βˆ’1)12𝑛sinπ‘›πœƒ=cosπ‘›πœƒβˆ’π‘›cos(π‘›βˆ’2)πœƒ+𝐢(𝑛,2)cos(π‘›βˆ’4)πœƒβˆ’πΆ(𝑛,3)cos(π‘›βˆ’6)πœƒ+β‹― 774 And when 𝑛 is odd, 2π‘›βˆ’1(βˆ’1)π‘›βˆ’12sinπ‘›πœƒ=sinπ‘›πœƒβˆ’π‘›sin(π‘›βˆ’2)πœƒ+𝐢(𝑛,2)sin(π‘›βˆ’4)πœƒβˆ’πΆ(𝑛,3)sin(π‘›βˆ’6)πœƒ+β‹― Observe that in these series the coefficients are those of the Binomial Theorem, with this exception: If 𝑛 be even, the last term must be divided by 2.
The series are obtained by expanding (π‘’π‘–πœƒΒ±π‘’βˆ’π‘–πœƒ)𝑛 by the Binomial Theorem, collecting the equidistant terms in pairs, and employing (768) and (769).

Expansion of cosπ‘›πœƒ and sinπ‘›πœƒ in powers of sinπœƒ

775 When 𝑛 is even, cosπ‘›πœƒ=1βˆ’π‘›22!sin2πœƒ+𝑛2(𝑛2βˆ’22)4!sin4πœƒβˆ’π‘›2(𝑛2βˆ’22)(𝑛2βˆ’42)6!sin6πœƒ+β‹― 776 When 𝑛 is odd, cosπ‘›πœƒ=cosπœƒ1βˆ’π‘›2βˆ’12!sin2πœƒ+(𝑛2βˆ’1)(𝑛2βˆ’32)4!sin4πœƒβˆ’(𝑛2βˆ’1)(𝑛2βˆ’32)(𝑛2βˆ’52)6!sin6πœƒ+β‹― 777 When 𝑛 is even, sinπ‘›πœƒ=cosπœƒsinπœƒβˆ’π‘›2βˆ’223!sin3πœƒ+(𝑛2βˆ’22)(𝑛2βˆ’42)5!sin5πœƒβˆ’(𝑛2βˆ’22)(𝑛2βˆ’42)(𝑛2βˆ’62)7!sin7πœƒ+β‹― 778 When 𝑛 is odd, sinπ‘›πœƒ=𝑛sinπœƒβˆ’π‘›(𝑛2βˆ’1)3!sin3πœƒ+𝑛(𝑛2βˆ’1)(𝑛2βˆ’32)5!sin5πœƒβˆ’π‘›(𝑛2βˆ’1)(𝑛2βˆ’32)(𝑛2βˆ’52)7!sin7πœƒ+β‹― Proof: By (758), we may assume, when 𝑛 is an even integer cosπ‘›πœƒ=1+𝐴2sin2πœƒ+𝐴4sin4πœƒ+β‹―+𝐴2π‘Ÿsin2π‘Ÿπœƒ+β‹― Put πœƒ+π‘₯ for πœƒ, and in cosπ‘›πœƒcos𝑛π‘₯βˆ’sinπ‘›πœƒsin𝑛π‘₯ substitute for cos𝑛π‘₯ and sin𝑛π‘₯ their values in powers of 𝑛π‘₯ from (764). Each term on the right is of the type 𝐴2π‘Ÿ(sinπœƒcosπ‘₯+cosπœƒsinπ‘₯)2π‘Ÿ. Make similar substitutions for cosπ‘₯ and sinπ‘₯ in powers of π‘₯. Collect the two coefficients of π‘₯2 in each term by the multinomial theorem (137) and equate them all to the coefficient of π‘₯2 on the left. In this equation write cos2πœƒ for 1βˆ’sin2πœƒ everywhere, and then equate the coefficients of sin2π‘Ÿπœƒ to obtain the relation between the successive equatities 𝐴2π‘Ÿ and 𝐴2π‘Ÿ+2 for the series (775).
When 𝑛 is an odd integer, begin by assuming, by (759) sinπ‘›πœƒ=𝐴1sinπœƒ+𝐴3sin3πœƒ+β‹― 779 The expansions of cosπ‘›πœƒ and sinπ‘›πœƒ in powers of cosπœƒ are obtained by changing πœƒ into 12πœ‹βˆ’πœƒ in (775) to (778).

Expansion of cosπ‘›πœƒ in descending powers of cosπœƒ

780 2cosπ‘›πœƒ=(2cosπœƒ)π‘›βˆ’π‘›(2cosπœƒ)π‘›βˆ’2+𝑛(π‘›βˆ’3)2!(2cosπœƒ)π‘›βˆ’4βˆ’β‹―+(βˆ’1)π‘Ÿπ‘›(π‘›βˆ’rβˆ’1)(π‘›βˆ’rβˆ’2)β‹―(π‘›βˆ’2r+1)r!(2cosπœƒ)π‘›βˆ’2r+β‹― up to the last positive power of 2cosπœƒ.
Proof: By expanding each term of the identity log(1βˆ’π‘₯𝑧)+log1βˆ’π‘§π‘₯=log1βˆ’π‘§π‘₯+1π‘₯βˆ’π‘§ By (156), equating coefficients of 𝑧𝑛, and substituting from (768). 783 sin𝛼+𝑐sin(𝛼+𝛽)+𝑐2sin(𝛼+2𝛽)+β‹― to 𝑛 terms =sinπ›Όβˆ’π‘sin(π›Όβˆ’π›½)βˆ’π‘π‘›sin(𝛼+𝑛𝛽)+𝑐𝑛+1sin{𝛼+(π‘›βˆ’1)𝛽}1βˆ’2𝑐cos𝛽+𝑐2 784 If 𝑐 be < 1 and 𝑛 infinite, this becomes =sinπ›Όβˆ’π‘sin(π›Όβˆ’π›½)1βˆ’2𝑐cos𝛽+𝑐2 785 cos𝛼+𝑐cos(𝛼+𝛽)+𝑐2cos(𝛼+2𝛽)+β‹― to 𝑛 terms = a similar result, changing sin into cos in the numerator. 786 similarly when 𝑐 is < 1 and 𝑛 infinite. 787 Method of summation: Substitute for the sines or cosines their exponential values (768). Sum the two resulting geometrical series, and substitute the sines or cosines again for the exponential values by (766). 788 𝑐sin(𝛼+𝛽)+𝑐22!sin(𝛼+2𝛽)+𝑐33!sin(𝛼+3𝛽)+β‹― to infinity =𝑒𝑐cos𝛽sin(𝛼+𝑐sin𝛽)βˆ’sin𝛼 789 𝑐cos(𝛼+𝛽)+𝑐22!cos(𝛼+2𝛽)+𝑐33!cos(𝛼+3𝛽)+β‹― to infinity =𝑒𝑐cos𝛽cos(𝛼+𝑐sin𝛽)βˆ’cos𝛼 Obtained by the rule in (787) 790 If, in the series (783) to (789), 𝛽 be changed into 𝛽+πœ‹, the signs of the alternate terms will thereby be changed.

Expansion of πœƒ in powers of tanπœƒ (Gregory's series)

791 πœƒ=tanπœƒβˆ’tan3πœƒ3+tan5πœƒ5βˆ’β‹― The series converges if tanπœƒ be not >1. Proof: By expanding the logarithm of the value of 𝑒2π‘–πœƒ in (771) by (158).

formula for the calculation of the value of πœ‹ by Gregor's series

792 πœ‹4=tanβˆ’112+tanβˆ’113=tanβˆ’115βˆ’tanβˆ’11239791 794 πœ‹4=4tanβˆ’115βˆ’tanβˆ’1170+tanβˆ’1199 Proof: By employing the formula for tan(𝐴±𝐡), (631)

To Prove that πœ‹ is Incommensurable

795 Convert the value of tanπœƒ in terms of πœƒ from (764) and (765) into a continued fraction, thus tanπœƒ=πœƒ1βˆ’πœƒ23βˆ’πœƒ25βˆ’πœƒ27βˆ’β‹―; or this result may be obtained by putting π‘–πœƒ for 𝑦 in (294), and by (770). Hence 1βˆ’πœƒtanπœƒ=πœƒ23βˆ’πœƒ25βˆ’πœƒ27βˆ’β‹― Put πœ‹2 for , and assume that πœ‹, and therefore πœ‹24, is commensurable. Let πœ‹24=π‘šπ‘›, π‘š and 𝑛 being integers. Then we shall have 1=π‘š2π‘›βˆ’π‘šπ‘›5π‘›βˆ’π‘šπ‘›7π‘›βˆ’β‹―
The continued fraction is incommensurable, by (177). But unity cannot be equal to an incommensurable quantity. Therefore πœ‹ is not commensurable. 796 If sinπ‘₯=𝑛sin(π‘₯+𝛼), π‘₯=𝑛sin𝛼+𝑛22sin2𝛼+𝑛33sin3𝛼+β‹― 797 If tanπ‘₯=𝑛tan𝑦, π‘₯=π‘¦βˆ’π‘šsin2𝑦+π‘š22sin4π‘¦βˆ’π‘š33sin6𝑦+β‹―, where π‘š=1βˆ’π‘›1+𝑛
Proof:By substiuting the exponential values of the sine or tangent (769) and (770), and then eliminating π‘₯. 798 Coefficient of π‘₯𝑛 in the expansion of π‘’π‘Žπ‘₯cos𝑏π‘₯=(π‘Ž2+𝑏2)𝑛2𝑛!cosπ‘›πœƒ, where π‘Ž=π‘Ÿcosπœƒ and 𝑏=π‘Ÿsinπœƒ.
For proof, substitute for cos𝑏π‘₯ from (768); expand by (150); put π‘Ž=π‘Ÿcosπœƒ and 𝑏=π‘Ÿsinπœƒ in the coefficient of 𝑒π‘₯, employ (757). 799 When 𝑒<1, 1βˆ’π‘’21βˆ’π‘’cosπœƒ=1+2𝑏cosπœƒ+2𝑏2cos2πœƒ+2𝑏3cos3πœƒ+β‹―, where 𝑏=𝑒1+1βˆ’π‘’2
For proof, put 𝑒=2𝑏1+𝑏2 and 2cosπœƒ=π‘₯+1π‘₯, expand the fraction in two series of powers of π‘₯ by the method of (257), and substitute from (768). 800 sin𝛼+sin(𝛼+𝛽)+sin(𝛼+2𝛽)+β‹―+sin{𝛼+(π‘›βˆ’1)𝛽}=sin𝛼+π‘›βˆ’12𝛽sin𝑛2𝛽sin𝛽2 801 cos𝛼+cos(𝛼+𝛽)+sin(𝛼+2𝛽)+β‹―+cos{𝛼+(π‘›βˆ’1)𝛽}=cos𝛼+π‘›βˆ’12𝛽sin𝑛2𝛽sin𝛽2 802 If the terms in these series have the signs + and βˆ’ alternately, change 𝛽 into 𝛽+πœ‹ in the results.
Proof: Multiply the series by 2sin𝛽2, and apply (669) and (666). 803 If 𝛽=2πœ‹π‘› in (800) and (801), each series vanishes. 804 Generally, if 𝛽=2πœ‹π‘›, and if π‘Ÿ be an integer not a multiple of 𝑛, the sum of the π‘Ÿth powers of the sines or cosines in (800) or (801) is zero if π‘Ÿ be odd; and if π‘Ÿ be even it is =𝑛2π‘Ÿ; by (772) to (774) 805 General Theorem: Denoting the sum of the series 𝑐+𝑐1π‘₯+𝑐2π‘₯2+β‹―+𝑐𝑛π‘₯𝑛 by 𝐹(π‘₯); then 𝑐cos𝛼+𝑐1cos(𝛼+𝛽)+β‹―+𝑐𝑛cos(𝛼+𝑛𝛽)=12{𝑒𝑖𝛼𝐹(𝑒𝑖𝛽)+π‘’βˆ’π‘–π›ΌπΉ(π‘’βˆ’π‘–π›½)} and 806 𝑐sin𝛼+𝑐1sin(𝛼+𝛽)+β‹―+𝑐𝑛sin(𝛼+𝑛𝛽)=12𝑖{𝑒𝑖𝛼𝐹(𝑒𝑖𝛽)βˆ’π‘’βˆ’π‘–π›ΌπΉ(π‘’βˆ’π‘–π›½)} Provd by substituting for the sines and cosines their exponential values (766), β‹―.

Expansion of the sine and cosine in factors

807 π‘₯2π‘›βˆ’2π‘₯𝑛𝑦𝑛cosπ‘›πœƒ+𝑦2𝑛=π‘₯2βˆ’2π‘₯𝑦cosπœƒ+𝑦2π‘₯2βˆ’2π‘₯𝑦cosπœƒ+2πœ‹π‘›+𝑦2β‹― to 𝑛 factors, adding 2πœ‹π‘› to the angle successively.
Proof: By solving the quadratic on the left, we get π‘₯=𝑦(cosπ‘›πœƒ+𝑖sinπ‘›πœƒ)1𝑛. The 𝑛 values of π‘₯ are found by (757) and (626), and thence tha factors. For the factors π‘₯𝑛±𝑦𝑛 see (480). 808 sinπ‘›πœ™=2π‘›βˆ’1sinπœ™sinπœ™+πœ‹π‘›sinπœ™+2πœ‹π‘›β‹― as far as 𝑛 factors of sines.
Proof: By putting π‘₯=𝑦=1 and πœƒ=2πœ™ in the last. 809 If 𝑛 be even, sinπ‘›πœ™=2π‘›βˆ’1sinπœ™cosπœ™sin2πœ‹π‘›βˆ’sin2πœ™sin22πœ‹π‘›βˆ’sin2πœ™β‹― 810 If 𝑛 be odd, omit cosπœ™ and make up 𝑛 factors, reckoning two factors for each pair of terms in brackets.
Proof: From (808), by collecting equidistant factors in pairs, and applying (659). 811 cosπ‘›πœ™=2π‘›βˆ’1sinπœ™+πœ‹2𝑛sinπœ™+3πœ‹2𝑛⋯ to 𝑛 factors. Proof: Put πœ™+πœ‹2𝑛 for πœ™ in (808). 812 Also, if 𝑛 be odd, cosπ‘›πœ™=2π‘›βˆ’1cosπœ™sin2πœ‹2π‘›βˆ’sin2πœ™sin23πœ‹2π‘›βˆ’sin2πœ™β‹― 813 If 𝑛 be even, omit cosπœ™,
Proof: as in (809) 814 𝑛=2π‘›βˆ’1sinπœ‹π‘›sin2πœ‹π‘›sin3πœ‹π‘›β‹―sin(π‘›βˆ’1)πœ‹π‘› Proof: divide (809) by sinπœ™, and make πœ™ vanish; then apply (754). 815 sinπœƒ=πœƒ1βˆ’πœƒπœ‹21βˆ’πœƒ2πœ‹21βˆ’πœƒ3πœ‹2β‹― 816 cosπœƒ=1βˆ’2πœƒπœ‹21βˆ’2πœƒ3πœ‹21βˆ’2πœƒ5πœ‹2β‹― Proof: Put πœ™=πœƒπ‘› in (809) and (842); divide by (814) and make 𝑛 infinite. 817 𝑒π‘₯βˆ’2cosπœƒ+π‘’βˆ’π‘₯=4sin2πœƒ21+π‘₯2πœƒ21+π‘₯2(2πœ‹Β±πœƒ)21+π‘₯2(4πœ‹Β±πœƒ)2β‹― Proved by substituting π‘₯=1+𝑧2𝑛, 𝑦=1βˆ’π‘§2𝑛, and πœƒπ‘› for πœƒ in (807) Making 𝑛 infinite, and reducing one series of factors to 4sin2πœƒ2 by putting 𝑧=0.

De Moivre's Property of the Circle

Circle: Take 𝑃 any point, and 𝑃𝑂𝐡=πœƒ any angle, 𝐡𝑂𝐢=𝐢𝑂𝐷=β‹―=2πœ‹π‘›; 𝑂𝑃=π‘₯; 𝑂𝐡=π‘Ÿ image 819 π‘₯2π‘›βˆ’2π‘₯π‘›π‘Ÿπ‘›cosπ‘›πœƒ+π‘Ÿ2𝑛=𝑃𝐡2𝑃𝐢2𝑃𝐷2β‹― to 𝑛 factors By (807) and (702), since 𝑃𝐡2=π‘₯2βˆ’2π‘₯π‘Ÿcosπœƒ+π‘Ÿ2, β‹― 820 If π‘₯=π‘Ÿ, 2π‘Ÿπ‘›sinπ‘›πœƒ2=𝑃𝐡⋅𝑃𝐢⋅𝑃𝐷⋯ 821

Cotes's Properties

If πœƒ=2πœ‹2, π‘₯π‘›βˆΌπ‘Ÿπ‘›=𝑃𝐡⋅𝑃𝐢⋅𝑃𝐷⋯ 822 π‘₯𝑛+π‘Ÿπ‘›=π‘ƒπ‘Žβ‹…π‘ƒπ‘β‹…π‘ƒπ‘β‹―

Sources and References


ID: 210900012 Last Updated: 9/12/2021 Revision: 0 Ref:



  1. B. Joseph, 1978, University Mathematics: A Textbook for Students of Science &amp; Engineering
  2. Ayres, F. JR, Moyer, R.E., 1999, Schaum's Outlines: Trigonometry
  3. Hopkings, W., 1833, Elements of Trigonometry

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