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Special Angles Kids: Sum and Difference Formulas

Samuel Dominic Chukwuemeka (SamDom For Peace) Special Angles Kids are those angles whose trigonometric functions can be found based on the trigonometric formulas of special angles.

Sum Formulas
(1.)sin(α+β)=sinαcosβ+cosαsinβ(2.)cos(α+β)=cosαcosβsinαsinβ(3.)tan(α+β)=tanα+tanβ1tanαtanβ Difference Formulas
(1.)sin(αβ)=sinαcosβcosαsinβ(2.)cos(αβ)=cosαcosβ+sinαsinβ(3.)tan(αβ)=tanαtanβ1+tanαtanβ Determine the trigonometric functions of these angles
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75
75=45+30
Sum Formula


sin75=sin(45+30)=sin45cos30+cos45sin30=2232+2212=2322+2122=64+24sin75=6+24


cos75=cos(45+30)=cos45cos30sin45sin30=22322212=23222122=6424cos75=624


tan75=tan(45+30)=tan45+tan301tan45tan30=(1+33)÷(1133)=(33+33)÷(133)=(3+33)÷(3333)=3+33÷333=3+33333=3+333=3+3333+33+3=(3+3)(3+3)(33)(3+3)=9+33+33+(3)232(3)2=9+63+393=12+636=6(2+3)6tan75=2+3


csc75=1sin75=1÷sin75=1÷6+24=146+2=46+26262=4(62)(6)2(2)2=4(62)62=4(62)4csc75=62


sec75=1cos75=1÷cos75=1÷624=1462=4626+26+2=4(6+2)(6)2(2)2=4(6+2)62=4(6+2)4sec75=6+2


cot75=1tan75=1÷tan75=1÷(2+3)=12+3=12+32323=23(2)2(3)2=2343=231cot75=23

15
15=4530
Difference Formula


sin15=sin(4530)=sin45cos30cos45sin30=22322212=23222122=6424sin15=624


cos15=cos(4530)=cos45cos30+sin45sin30=2232+2212=2322+2122=64+24cos15=6+24


tan15=tan(4530)=tan45tan301+tan45tan30=(133)÷(1+133)=(3333)÷(1+33)=(333)÷(33+33)=333÷3+33=33333+3=333+3=333+33333=(33)(33)(3+3)(33)=93333+(3)232(3)2=963+393=12636=6(23)6tan15=23


csc15=1sin15=1÷sin15=1÷624=1462=4626+26+2=4(6+2)(6)2(2)2=4(6+2)62=4(6+2)4csc15=6+2


sec15=1cos15=1÷cos15=1÷6+24=146+2=46+26262=4(62)(6)2(2)2=4(62)62=4(62)4sec15=62


cot15=1tan15=1÷tan15=1÷(23)=123=1232+32+3=2+3(2)2(3)2=2+343=2+31cot15=2+3

105
105=13530
Difference Formula


sin105=sin(13530)=sin135cos30cos135sin30=22322212=2322+2122=64+24=24+64sin105=2+64


cos105=cos(13530)=cos135cos30+sin135sin30=2232+2212=21222322=2464cos105=264


tan105=tan(13530)=tan135tan301+tan135tan30=(133)÷(1+133)=(3333)÷(133)=(333)÷(3333)=333÷333=333333=3333=33333+33+3=(33)(3+3)(33)(3+3)=93333(3)232(3)2=963393=12636=6(23)6tan105=23


csc105=1sin105=1÷sin105=1÷6+24=146+2=46+26262=4(62)(6)2(2)2=4(62)62=4(62)4csc105=62


sec105=1cos105=1÷cos105=1÷264=1426=4262+62+6=4(2+6)(2)2(6)2=4(2+6)26=4(2+6)4=1(2+6)sec105=26


cot105=1tan105=1÷tan105=1÷(23)=123=1232+32+3=2+3(2)2(3)2=2+343=2+31=2+3cot105=32

165
165=135+30
Sum Formula


sin165=sin(135+30)=sin135cos30+cos135sin30=2232+2212=23222122=6424sin165=624


cos165=cos(135+30)=cos135cos30sin135sin30=22322212=23222122=6424cos165=624


tan165=tan(135+30)=tan135+tan301tan135tan30=(1+33)÷(1133)=(33+33)÷(1+33)=(3+33)÷(33+33)=3+33÷3+33=3+3333+3=3+33+3=3+33+33333=(3+3)(33)(3+3)(33)=9+33+33(3)232(3)2=9+63393=12+636=6(2+3)6=2+3tan165=32


csc165=1sin165=1÷sin165=1÷624=1462=4626+26+2=4(6+2)(6)2(2)2=4(6+2)62=4(6+2)4csc165=6+2


sec165=1cos165=1÷cos165=1÷624=1462=4626+26+2=4(6+2)(6)2(2)2=4(6+2)62=4(6+2)4=6+2sec165=26


cot165=1tan165=1÷tan165=1÷(2+3)=12+3=12+32323=23(2)2(3)2=2343=231cot165=23





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255
255=210+45
Sum Formula


sin255=sin(210+45)=sin210cos45+cos210sin45=1222+3222=12223222=2464sin255=264


cos255=cos(210+45)=cos210cos45sin210sin45=32221222=3222+1222=64+24cos255=264


tan255=tan(210+45)=tan210+tan451tan210tan45=(33+1)÷(1331)=(33+33)÷(133)=(3+33)÷(3333)=3+33÷333=3+33333=3+333=3+3333+33+3=33+(3)2+9+3332(3)2=33+3+9+3393=12+636=6(2+3)6=2+3tan255=2+3


csc255=1sin255=1÷sin255=1÷264=1426=4262+62+6=4(2+6)(2)2(6)2=4(2+6)26=4(2+6)4=1(2+6)=26csc255=26


sec255=1cos255=1÷cos255=1÷264=1426=4262+62+6=4(2+6)(2)2(6)2=4(2+6)26=4(2+6)4=1(2+6)=26sec255=26


cot255=1tan255=1÷tan255=1÷(2+3)=12+3=12+32323=23(2)2(3)2=2343=231cot255=23

195
195=22530
Difference Formula


sin195=sin(22530)=sin225cos30cos225sin30=22322212=2322+2122=64+24=2464sin195=264


cos195=cos(22530)=cos225cos30+sin225sin30=2232+2212=23222122=6424cos195=624


tan195=tan(22530)=tan225tan301+tan225tan30=(133)÷(1+133)=(3333)÷(1+33)=(333)÷(33+33)=333÷3+33=33333+3=333+3=333+33333=(33)(33)(3+3)(33)=93333+(3)232(3)2=963+393=12636=6(23)6tan195=23


csc195=1sin195=1÷sin195=1÷264=1426=4262+62+6=4(2+6)(2)2(6)2=4(2+6)26=4(2+6)4=1(2+6)csc195=26


sec195=1cos195=1÷cos195=1÷624=1462=4626+26+2=4(6+2)(6)2(2)2=4(6+2)62=4(6+2)4=6+2sec195=26


cot195=1tan195=1÷tan195=1÷(23)=123=1232+32+3=2+3(2)2(3)2=2+343=2+31cot195=2+3

285
285=33045
Difference Formula


sin285=sin(33045)=sin330cos45cos330sin45=12223222=12223222=2464sin285=264


cos285=cos(33045)=cos330cos45+sin330sin45=3222+1222=32221222=6424cos285=624


tan285=tan(33045)=tan330tan451+tan330tan45=(331)÷(1+331)=(3333)÷(133)=(333)÷(3333)=333÷333=333333=3333=33333+33+3=(33)(3+3)(33)(3+3)=33(3)293332(3)2=396393=12636=6(23)6tan285=23


csc285=1sin285=1÷sin285=1÷264=1426=4262+62+6=4(2+6)(2)2(6)2=4(2+6)26=4(2+6)4=1(2+6)=26csc285=26


sec285=1cos285=1÷cos285=1÷624=1462=4626+26+2=4(6+2)(6)2(2)2=4(6+2)62=4(6+24sec285=6+2


cot285=1tan285=1÷tan285=1÷(23)=123=1232+32+3=2+3(2)2(3)2=2+343=2+31=2+3cot285=32

345
345=300+45
Sum Formula


sin345=sin(300+45)=sin300cos45+cos300sin45=3222+1222=3222+1222=64+24=2464sin345=264


cos345=cos(300+45)=cos300cos45sin300sin45=12223222=1222+3222=24+64cos345=2+64


tan345=tan(300+45)=tan300+tan451tan300tan45=3+11(3)(1)=1313=131+3=131+31313=(13)(13)(1+3)(13)=133+(3)212(3)2=123+313=4232=2(23)2=(23)=2+3tan345=32


csc345=1sin345=1÷sin345=1÷264=1426=4262+62+6=4(2+6)(2)2(6)2=4(2+6)26=4(2+6)4=(2+6)=26csc345=26


sec345=1cos345=1÷cos345=1÷2+64=142+6=42+62626=4(26)(2)2(6)2=4(26)26=4(26)4=(26)=2+6sec345=62


cot345=1tan345=1÷tan345=1÷(32)=132=1323+23+2=3+2(3)2(2)2=3+234=3+21=(3+2)=32cot345=23





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