Angle yana auna juyawa ko karkata tsakanin layuka biyu. Yi tunanin buɗe kofa ko juya dabaran. Ana auna shi a digiri (°), radian (rad), ko gradian. 360° = cikakken da'ira = juyawa ɗaya cikakke.

• Angle = adadin juyawa
• Cikakken da'ira = 360° = 2π rad
• Kusurwa madaidaiciya = 90° = π/2 rad
• Layi madaidaici = 180° = π rad

Digiri: da'ira da aka raba zuwa sassa 360 (tarihi). Radian: dangane da radius na da'irar. Radian 2π = 360°. Radian suna da 'halitta' don lissafi/kimiyyar lissafi. π rad = 180°, don haka 1 rad ≈ 57.3°.

• 360° = 2π rad (cikakken da'ira)
• 180° = π rad (rabin da'ira)
• 90° = π/2 rad (kusurwa madaidaiciya)
• 1 rad ≈ 57.2958° (canji)

Gradian: 100 grad = 90° (kusurwar awo). Mintin/sakan baka: rabe-raben digiri (ilimin taurari). Mil: kewayawar soja (mil 6400 = da'ira). Kowace raka'a don takamaiman aikace-aikace.

• Gradian: 400 grad = da'ira
• Mintin baka: 1′ = 1/60°
• Sakan baka: 1″ = 1/3600°
• Mil (NATO): 6400 mil = da'ira

360° a kowace da'ira (asalin Babila - ~360 kwana/shekara). An raba shi: 1° = 60′ (mintuna na baka) = 3600″ (dakiku na baka). Na duniya don kewayawa, bincike, amfani da yau da kullun.

• 360° = cikakken da'ira
• 1° = 60 mintuna na baka (′)
• 1′ = 60 dakiku na baka (″)
• Sauƙi ga mutane, tarihi

Radian: tsawon baka = radius. 2π rad = kewaye da da'ira/radius. Na halitta don lissafi (abubuwan da aka samo na sin, cos). Matsayin kimiyyar lissafi, injiniyanci. π rad = 180°.

• 2π rad = 360° (daidai)
• π rad = 180°
• 1 rad ≈ 57.2958°
• Na halitta don lissafi/kimiyyar lissafi

Gradian: 400 grad = da'ira (kusurwar awo). 100 grad = kusurwa madaidaiciya. Mil: kewayawar soja - NATO na amfani da mil 6400. USSR ta yi amfani da 6000. Akwai ma'auni daban-daban.

• 400 grad = 360°
• 100 grad = 90° (kusurwa madaidaiciya)
• Mil (NATO): 6400 a kowace da'ira
• Mil (USSR): 6000 a kowace da'ira

rad = deg × π/180. deg = rad × 180/π. grad = deg × 10/9. Koyaushe yi amfani da radian a cikin lissafi! Ayyukan trig suna buƙatar radian don abubuwan da aka samo.

• rad = deg × (π/180)
• deg = rad × (180/π)
• grad = deg × (10/9)
• Lissafi yana buƙatar radian

sin, cos, tan suna danganta kusurwoyi da rabo. Da'irar raka'a: radius=1, kusurwa=θ. Matsayin maki: (cos θ, sin θ). Mahimmanci don kimiyyar lissafi, injiniyanci, zane-zane.

• sin θ = gaba/hypotenuse
• cos θ = kusa/hypotenuse
• tan θ = gaba/kusa
• Da'irar raka'a: (cos θ, sin θ)

Ana ƙara/rage kusurwoyi kamar yadda aka saba. 45° + 45° = 90°. Cikakken juyawa: ƙara/rage 360° (ko 2π). Lissafin modulo don nannade: 370° = 10°.

• θ₁ + θ₂ (ƙari na yau da kullun)
• Nannade: θ mod 360°
• 370° ≡ 10° (mod 360°)
• Kusurwoyi marasa kyau: -90° = 270°

Alamar kompas: 0°=Arewa, 90°=Gabas, 180°=Kudu, 270°=Yamma. Sojoji suna amfani da mil don daidaito. Kompas yana da maki 32 (11.25° kowanne). GPS yana amfani da digiri na goma.

• Alama: 0-360° daga Arewa
• NATO mil: 6400 a kowace da'ira
• Maki na kompas: 32 (11.25° kowanne)
• GPS: digiri na goma

Matsayin taurari: daidaiton sakan baka. Parallax: millarcseconds. Hubble: ~50 mas ƙuduri. Tauraron dan adam na Gaia: daidaiton microarcsecond. Kusurwar sa'a: 24h = 360°.

• Sakan baka: matsayin taurari
• Millarcsecond: parallax, VLBI
• Microarcsecond: tauraron dan adam na Gaia
• Kusurwar sa'a: 15°/awa

Karkata: kashi na daraja ko kusurwa. 10% daraja ≈ 5.7°. Zane-zanen hanya yana amfani da kashi. Bincike yana amfani da digiri/mintuna/dakiku. Tsarin Gradian don ƙasashen awo.

• Karkata: % ko digiri
• 10% ≈ 5.7° (arctan 0.1)
• Bincike: DMS (deg-min-sec)
• Gradian: binciken awo

rad = deg × π/180. deg = rad × 180/π. Mai sauri: 180° = π rad, don haka raba/ninka da wannan rabo.

• rad = deg × 0.01745
• deg = rad × 57.2958
• π rad = 180° (daidai)
• 2π rad = 360° (daidai)

kusurwa = arctan(karkata/100). 10% karkata = arctan(0.1) ≈ 5.71°. Akasin: karkata = tan(kusurwa) × 100.

• θ = arctan(daraja/100)
• 10% → arctan(0.1) = 5.71°
• 45° → tan(45°) = 100%
• Mai tsanani: 100% = 45°

1° = 60′ (mintin baka). 1′ = 60″ (sakan baka). Jimla: 1° = 3600″. Rarraba mai sauri don daidaito.

• 1° = 60 mintuna na baka
• 1′ = 60 dakiku na baka
• 1° = 3600 dakiku na baka
• DMS: digiri-mintuna-dakiku

Hanya tana da darajar 8%. Menene kusurwar?

θ = arctan(8/100) = arctan(0.08) ≈ 4.57°. Karkata mai sauƙi!

Kewaya alamar 135°. Wace hanya ce ta kompas?

0°=N, 90°=E, 180°=S, 270°=W. 135° yana tsakanin E (90°) da S (180°). Hanyar: Kudu maso Gabas (SE).

Tauraro ya motsa sakan baka 0.5. Digiri nawa ne?

1″ = 1/3600°. Don haka 0.5″ = 0.5/3600 = 0.000139°. Motsi kaɗan!

Babila sun yi amfani da tsarin tushe-60 (sexagesimal). 360 yana da masu rarraba da yawa (abubuwa 24!). Kimanin yayi daidai da kwanaki 360 a cikin shekara. Mai dacewa don ilimin taurari da kiyaye lokaci. Hakanan yana rarraba daidai da 2, 3, 4, 5, 6, 8, 9, 10, 12...

An ayyana radian ta tsawon baka = radius. Yana sa lissafi ya zama kyakkyawa: d/dx(sin x) = cos x (kawai a cikin radian!). A digiri, d/dx(sin x) = (π/180)cos x (rikitarwa). Halitta 'tana amfani' da radian!

Kusurwar awo: 100 grad = kusurwa madaidaiciya. An gwada lokacin Juyin Juya Halin Faransa da tsarin awo. Bai taɓa zama sananne ba—digiri sun yi zurfi sosai. Har yanzu ana amfani da shi a wasu bincike (Switzerland, arewacin Turai). Na'urorin lissafi suna da yanayin 'grad'!

1 millarcsecond ≈ faɗin gashin ɗan adam da aka gani daga nisan kilomita 10! Na'urar hangen nesa ta Hubble na iya warware ~50 mas. Daidaito mai ban mamaki don ilimin taurari. Ana amfani da shi don auna parallax na taurari, taurari biyu.

Mil na soja: 1 mil ≈ faɗin mita 1 a nisan kilomita 1 (NATO: 1.02 m, kusa da isa). Lissafin hankali mai sauƙi don kimanta nisa. Ƙasashe daban-daban suna amfani da mil daban-daban (6000, 6300, 6400 a kowace da'ira). Raka'ar kimiyyar harsasai mai amfani!

90 = 360/4 (juyawa kwata). Amma 'madaidaiciya' ta fito daga Latin 'rectus' = a tsaye, madaidaiciya. Kusurwa madaidaiciya tana yin layuka masu tsaye. Mahimmanci don gini—gine-gine suna buƙatar kusurwoyi madaidaiciya don tsayawa!

Babila sun yi amfani da tsarin lamba na sexagesimal (tushe-60) don ilimin taurari da kiyaye lokaci. Sun raba da'irar zuwa sassa 360 saboda 360 ≈ kwana a cikin shekara (ainihin 365.25), kuma 360 yana da masu rarraba 24—mai matuƙar dacewa don ɓangarori.

Wannan tsarin tushe-60 ya ci gaba har yau: sakan 60 a cikin minti, mintuna 60 a cikin awa da kuma a cikin digiri. Lambar 360 tana da abubuwan da suka haɗa da 2³ × 3² × 5, tana rarraba daidai da 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180—mafarkin na'urar lissafi!

• 2000 KZ: Masana ilimin taurari na Babila suna bin diddigin matsayin sararin samaniya a digiri
• An zaɓi 360° don rarrabawa da kimanin shekara
• Tushe-60 yana ba mu sa'o'i (24 = 360/15) da mintuna/dakiku
• Masana ilimin taurari na Girka sun karɓi 360° daga teburan Babila

Abubuwan Euclid (300 KZ) sun tsara ilimin lissafin kusurwa—kusurwoyi madaidaiciya (90°), masu cikawa (jimla zuwa 90°), masu ƙari (jimla zuwa 180°). Masana lissafi na Girka kamar Hipparchus sun ƙirƙiri trigonometry ta amfani da teburan da suka dogara da digiri don ilimin taurari da bincike.

Masu kewayawa na zamani na tsakiya sun yi amfani da astrolabe da kompas tare da maki 32 (kowanne 11.25°). Jiragen ruwa suna buƙatar daidaitattun alamomi; mintuna na baka (1/60°) da dakiku na baka (1/3600°) sun fito don kasidun taurari da taswirorin ruwa.

• 300 KZ: Abubuwan Euclid sun bayyana kusurwoyin lissafi
• 150 KZ: Hipparchus ya ƙirƙiri teburan trig na farko (digiri)
• 1200s: Astrolabe yana amfani da alamun digiri don kewayawar sararin samaniya
• 1569: Tsarin taswirar Mercator yana buƙatar lissafin da ke kiyaye kusurwa

Yayin da Newton da Leibniz suka haɓaka lissafi (1670s), digiri sun zama matsala: d/dx(sin x) = (π/180)cos x a digiri—lamba mai muni! Roger Cotes (1682-1716) da Leonhard Euler sun tsara radian: kusurwa = tsawon baka / radius. Yanzu d/dx(sin x) = cos x da kyau.

James Thomson ya ƙirƙiri kalmar 'radian' a cikin 1873 (daga Latin 'radius'). Radian ya zama raka'ar don nazarin lissafi, kimiyyar lissafi, da injiniyanci. Duk da haka digiri sun ci gaba a rayuwar yau da kullun saboda mutane sun fi son lambobi masu yawa fiye da π.

• 1670s: Lissafi ya bayyana cewa digiri suna haifar da dabaru masu rikitarwa
• 1714: Roger Cotes ya haɓaka 'awon madauwari' (pre-radian)
• 1748: Euler yana amfani da radian sosai a cikin nazari
• 1873: Thomson ya ba shi suna 'radian'; ya zama ma'aunin lissafi

Manyan makamai na Yaƙin Duniya na I sun buƙaci raka'o'in kusurwa masu amfani: an haifi mil—mil 1 ≈ karkacewar mita 1 a nisan kilomita 1. NATO ta daidaita mil 6400/da'ira (lamba mai kyau na 2), yayin da USSR ta yi amfani da 6000 (sauƙin goma). Gaskiyar milliradian = 6283/da'ira.

Ilimin taurari na zamanin sararin samaniya ya sami daidaiton millarcsecond (Hipparcos, 1989), sannan microarcseconds (Gaia, 2013). Gaia tana auna parallax na taurari zuwa microarcseconds 20—daidai da ganin gashin ɗan adam daga nisan kilomita 1,000! Kimiyyar lissafi ta zamani tana amfani da radian a duniya; kawai kewayawa da gini ne har yanzu ke son digiri.

• 1916: Manyan makamai na soja sun karɓi mil don lissafin nisa
• 1960: SI ta amince da radian a matsayin raka'ar da aka samo mai haɗin kai
• 1989: Tauraron dan adam na Hipparcos: ~1 millarcsecond daidaito
• 2013: Tauraron dan adam na Gaia: daidaiton microarcsecond 20—yana taswirar taurari biliyan 1

Yi amfani da digiri don: kusurwoyin yau da kullun, kewayawa, bincike, gini. Yi amfani da radian don: lissafi, daidaitattun kimiyyar lissafi, shirye-shirye (ayyukan trig). Radian suna da 'halitta' saboda tsawon baka = radius × kusurwa. Abubuwan da aka samo kamar d/dx(sin x) = cos x suna aiki ne kawai a cikin radian!

Kewaye da da'ira = 2πr. Rabin da'ira (layi madaidaici) = πr. An ayyana radian a matsayin tsawon baka/radius. Don rabin da'ira: baka = πr, radius = r, don haka kusurwa = πr/r = π radian. Saboda haka, π rad = 180° ta ma'ana.

Yi amfani da arctan: kusurwa = arctan(daraja/100). Misali: 10% daraja = arctan(0.1) ≈ 5.71°. KADA ka ninka kawai! 10% ≠ 10°. Akasin: daraja = tan(kusurwa) × 100. 45° = tan(45°) × 100 = 100% daraja.

Mintin baka (′) = 1/60 na digiri (kusurwa). Mintin lokaci = 1/60 na awa (lokaci). Daban-daban gaba ɗaya! A ilimin taurari, 'mintin lokaci' yana canzawa zuwa kusurwa: minti 1 = mintuna 15 na baka (saboda 24h = 360°, don haka minti 1 = 360°/1440 = 0.25° = 15′).

An tsara mil don 1 mil ≈ mita 1 a kilomita 1 (kimiyyar harsasai mai amfani). Gaskiyar milliradian na lissafi = 1/1000 rad ≈ 6283 a kowace da'ira. NATO ta sauƙaƙa zuwa 6400 (ikon 2, yana rarraba da kyau). USSR ta yi amfani da 6000 (yana rarraba da 10). Sweden 6300 (sulhu). Dukkansu suna kusa da 2π×1000.

Ee! Mai kyau = agogon agogo (yarjejeniyar lissafi). Mara kyau = agogon agogo. -90° = 270° (matsayi ɗaya, hanya daban). A kewayawa, yi amfani da kewayon 0-360°. A lissafi/kimiyyar lissafi, kusurwoyi marasa kyau sun zama ruwan dare. Misali: -π/2 = -90° = 270°.