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Trigonometric ratios: • Trigonometric Ratios | Sine | Cosine | Tan...","preview":{"posterSrc":"//avatars.mds.yandex.net/get-vthumb/966322/505490c25d99ff6a482227c4dcf89199/564x318_1"},"target":"_self","position":"15","reqid":"1769720205505103-17304330184308580692-balancer-l7leveler-kubr-yp-klg-320-BAL","isAdultDoc":false,"relatedParams":{"text":"Trigonometric formulas | Relation among Trigonometric ratios","related_orig_text":"Trigonometric","related_porno":false,"related_less_3m_off":true,"client":"d2d","no_cnt":1,"related_src":"serp","related":"{\"porno\":false,\"vfp\":1,\"orig_text\":\"Trigonometric\",\"url\":\"http:\\/\\/www.youtube.com\\/watch?v=1hN24uFXGwc\",\"src\":\"serp\",\"rvb\":\"Eq8DChQxNDY0NjA5MjM4ODUxMDU3NDY2MgoUMTc2ODgzNzMxOTg0Mzg1NjcwNjUKEzY2NjY0ODY2NjI5ODA1NTU1NjUKFDE0NTA2MDQyNzM1NzEwNzM2OTg4ChQxMDUzOTU5OTE0NTIyODM5MjgwNwoUMTA3MTI0NjkwMDg2NDAwNTEyNDMKEzMyOTExMzg4NjQxODMxNDAzOTkKEzgwMTU0ODA2NDIzODIyOTgzOTYKFDE1MDA2Nzk4OTQ2NDcyMjQ5MTk5ChM5NDE1NzI5ODA4OTk0OTU0OTc3ChM4ODA5NjAyMTYxMjMyNDc5MzYwChQxNDEyODQ3MDYzMjUwNzUyNzM2NgoTNzU5NjI1NTk5MjQyNzc4Mzg4MwoTNzI4NDUwNTI5OTMwOTE2OTM4MQoUMTIwMzQ3Mzk4NDI0Mzc5MDI4NTcKFDE2ODk1NTM4OTY4OTYyNDEwNzQzChQxNjE2MDY5NDczNTYzNjY0MDk4NwoUMTIxMTc5NjY0NTQyNDg5Mjc1MjUKEzk5MDIyMTgzMTM5MDE2ODUxMzkKEzU5Mjk4Njk1Mjk5OTE2NDc0NDAaFQoTNzI4NDUwNTI5OTMwOTE2OTM4MVoTNzI4NDUwNTI5OTMwOTE2OTM4MWqvDRIBMBgAIkUaMQAKKmhodWdqbWZlanVqd21md2JoaFVDNk1DMlVKTDBlRkpNdjRXd2VwNFd5URICABIqEMIPDxoPPxPqCoIEJAGABCsqiwEQARp4gfb0BQX9AwDxB_0JAAT-AQEM7__3_v0A9vr7_f4C_wD68xMCBAAAAP8GAf8FAAAA9w8C8f__AAAK9gAB-gAAAAkC-AD9AAAAEvoG9f8BAAD2-gb5AgAAAAv2AQYAAAAA_AL_7gAAAAAH_PwGAAAAAAEEBv7_AAAAIAAtSprcOzgTQAlITlACKnMQABpg_xUACBH4zu4ZAfL34fcJ_uUe4QrU-wD-8wAeJPz_Lej05xARAB30AQbRAAAAEvj_DCIA_DX94gAVDwQCzN4BGxR_Bwz7HvwC_83vDCImG_4G8gr6AAn-C_0cBgoGHi4aIAAtzPeEOzgTQAlIb1ACKq8GEAwaoAYAAABCAACAQAAAAMAAANjBAAD-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-EEAAFDBAAAMwgAA4EEAAGBBAABQQQAAsEEAAChCAACYwQAAgMEAAJhBAAAoQgAAikIAAIC_AACgQQAA8MEAAHDCAACcQgAAYEIAAOBAAAAUwgAAUMEAAITCAAAwwgAAAEAAAABBAACwwQAA6MEAAIRCAAAAwQAA4EAAAEBAAAAcwgAA-MEAAAxCAAA0QgAA4EAAAJhCAABgQQAA4EAAAFDBAABgwgAAHEIAABjCAABkQgAAwMAAAOBBAABAQgAAdMIAAGxCAAC4QQAA2EEAALDBAABAwAAAgEAAAMBAAACIQQAAEMEAAIjBAAAMwgAAjMIAAKDBAAAcwgAAAMEAADTCAADQwQAAqMEAANhBAACAwgAAeEIAAARCAADgwAAAiEEAAODBAABEQgAAwMEAAIjBAACYwQAAREIAAHDBAABwQgAAaMIAAOjBAAAAQAAAkMEAAODBAABYQgAAZMIAAFDBAAA4wgAAbEIAACDBAACoQQAAIMIAAARCAADIQQAA-EEAACBCAAAowgAASEIAAIBBAAAQQiAAOBNACUh1UAEqjwIQABqAAgAAoDwAAKi9AACmPgAAyL0AAEy-AAB8PgAAbD4AABm_AAAwvQAAiL0AACy-AAA0vgAAhj4AAJo-AACovQAAQLwAAOg9AADgPAAALD4AABM_AAB_PwAAhr4AAAQ-AAAsPgAAoDwAAJg9AAC4PQAAgLsAAGw-AAD4PQAA6D0AAL6-AAAwvQAALL4AAFw-AAAMvgAAFL4AAGS-AACevgAAcL0AAHC9AAAkvgAAFD4AAGS-AABcvgAABL4AAJY-AABcvgAA-L0AABy-AABEvgAAUL0AAKY-AACAOwAADL4AAOg9AABVPwAAEL0AAIg9AACAuwAA4LwAAPg9AACYPQAALL4gADgTQAlIfFABKo8CEAEagAIAAIC7AABQPQAA-L0AAEW_AABMvgAAUD0AAAQ-AACYvQAALL4AAAw-AAAQPQAABL4AADA9AAC4vQAAUD0AAJi9AABEvgAAGT8AAIg9AACuPgAAcD0AAAS-AABAPAAA6L0AACy-AACKvgAAQLwAAEC8AABAvAAAQDwAAAw-AAAwPQAA6L0AAIg9AAAwPQAAHL4AAKg9AACePgAAor4AAOi9AAC4PQAAbD4AADA9AABAvAAAgDsAACw-AAB_vwAA-D0AAMo-AAAcvgAATD4AANi9AADoPQAAPD4AAPg9AAD4PQAA4DwAAJg9AABwPQAAEL0AAJg9AADYPQAA2L0AAAy-IAA4E0AJSHxQATAJOAFKAGAAaAA,\"}","related_url":"http://www.youtube.com/watch?v=1hN24uFXGwc","parent-reqid":"1769720205505103-17304330184308580692-balancer-l7leveler-kubr-yp-klg-320-BAL","related_vfp":1,"relatedVideo":"yes"},"cwidth":1280,"cheight":720,"cratio":1.77777,"dups":["7284505299309169381"],"episode":0,"season":0,"isEmbedOnly":false,"greenHost":"YouTube","hasTranslation":true,"contentTypeId":null,"censored":false},"12034739842437902857":{"videoId":"12034739842437902857","docid":"34-5-7-ZC580B7359472A5D2","description":"With Guided Math, we're completing the 11th Grade Math topics with MatBook 11. 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Öncelikle bir fonksiyonun tersinin olabilmesi için birebir ve örten olması gerektiği hatırlatılmakta, ardından sinüs, kosinüs ve tanjant fonksiyonlarının terslerinin nasıl alınacağı, tanım ve değer kümeleri, birim çember üzerindeki yerleri ve özellikleri açıklanmaktadır.","Ders boyunca grafik tableti kullanılarak konular görsel olarak desteklenmekte ve 14 farklı örnek soru çözülmektedir. Ayrıca radyan ve derece cinsinden değer bulma yöntemleri, trigonometrik ifadelerin sadeleştirilmesi ve fonksiyonların terslerinin nasıl yazılacağı adım adım gösterilmektedir. Video, bir sonraki dersin (17. trigonometri dersi) hakkında bilgi vererek ve geometri kampı hakkında tavsiye ile sonlanmaktadır."]},"endTime":3249,"title":"11. Sınıf Trigonometri Dersi: Ters Trigonometrik Fonksiyonlar","beginTime":0}],"fullResult":[{"index":0,"title":"Dersin Tanıtımı","list":{"type":"unordered","items":["Sercan Hoca, 11. sınıf tayfasına selamlaşıyor ve Mehmet Hoca'nın izin verdiği bir dersi anlatacağını belirtiyor.","Mehmet Hoca'nın video çekimleri ve İstanbul'a gitmesi nedeniyle dersi Sercan Hoca anlatacak.","Ders 17. derste trigonometri konusunun ters trigonometrik fonksiyonlara geçişinde olacak."]},"beginTime":0,"endTime":67,"href":"/video/preview/12034739842437902857?parent-reqid=1769720205505103-17304330184308580692-balancer-l7leveler-kubr-yp-klg-320-BAL&text=Trigonometric&t=0&ask_summarization=1"},{"index":1,"title":"Ters Fonksiyonların Önemi","list":{"type":"unordered","items":["Ters fonksiyonların önemini hatırlatmak için adım adım öğretim yapılacak.","Bir fonksiyonun ters fonksiyon olabilmesi için birebir ve örten olması gerekiyor."]},"beginTime":67,"endTime":119,"href":"/video/preview/12034739842437902857?parent-reqid=1769720205505103-17304330184308580692-balancer-l7leveler-kubr-yp-klg-320-BAL&text=Trigonometric&t=67&ask_summarization=1"},{"index":2,"title":"Birebir Fonksiyon","list":{"type":"unordered","items":["Birebir fonksiyon, tanım kümesindeki her elemanın görüntü kümesinde kendine ait bir görüntüye sahip olması durumunda oluşur.","Her tanım kümesi elemanının kendine ait bir görüntüsü varsa, bu fonksiyona birebir fonksiyon denir."]},"beginTime":119,"endTime":189,"href":"/video/preview/12034739842437902857?parent-reqid=1769720205505103-17304330184308580692-balancer-l7leveler-kubr-yp-klg-320-BAL&text=Trigonometric&t=119&ask_summarization=1"},{"index":3,"title":"Örten Fonksiyon","list":{"type":"unordered","items":["Örten fonksiyon, değer kümesinde açıkta eleman kalmaması durumunda oluşur.","Değer kümesindeki bütün elemanları doldurabiliyorsanız, fonksiyon örten fonksiyondur."]},"beginTime":189,"endTime":229,"href":"/video/preview/12034739842437902857?parent-reqid=1769720205505103-17304330184308580692-balancer-l7leveler-kubr-yp-klg-320-BAL&text=Trigonometric&t=189&ask_summarization=1"},{"index":4,"title":"Birebir ve Örten Fonksiyonların Grafiksel İncelenmesi","list":{"type":"unordered","items":["Birebirliği incelemek için yatay doğru testi yapılır; x eksenine paralel doğrular grafiği tek bir noktada kesiyorsa fonksiyon birebirdir.","Örtenliği incelemek için grafiğin değer kümesini doldurup doldurmadığı kontrol edilir; değer kümesindeki tüm elemanlar grafiğe ait bir görüntüye sahipse fonksiyon örten fonksiyondur.","Örnek grafiklerle birebir ve örten fonksiyonların özellikleri gösterilmiştir."]},"beginTime":229,"endTime":401,"href":"/video/preview/12034739842437902857?parent-reqid=1769720205505103-17304330184308580692-balancer-l7leveler-kubr-yp-klg-320-BAL&text=Trigonometric&t=229&ask_summarization=1"},{"index":5,"title":"Fonksiyonların Tersi İçin Gerekli Koşullar","list":{"type":"unordered","items":["Bir fonksiyonun tersinin olabilmesi için birebir ve örten olması gerekir.","Bir fonksiyon birebir ve örten olduğunda tersi vardır, aksi takdirde tersi yoktur."]},"beginTime":403,"endTime":444,"href":"/video/preview/12034739842437902857?parent-reqid=1769720205505103-17304330184308580692-balancer-l7leveler-kubr-yp-klg-320-BAL&text=Trigonometric&t=403&ask_summarization=1"},{"index":6,"title":"Sinüs Fonksiyonunun Grafiği","list":{"type":"unordered","items":["Sinüs fonksiyonunun grafiği x-y koordinat sisteminde çizildiğinde, -1 ile 1 aralığında zigzaglar yaparak gider.","Sinüs fonksiyonunun değerleri 0, π/2, π, 3π/2, 2π şeklinde döngüsel olarak değişir."]},"beginTime":444,"endTime":530,"href":"/video/preview/12034739842437902857?parent-reqid=1769720205505103-17304330184308580692-balancer-l7leveler-kubr-yp-klg-320-BAL&text=Trigonometric&t=444&ask_summarization=1"},{"index":7,"title":"Sinüs Fonksiyonunun Tersinin Varlığı","list":{"type":"unordered","items":["Sinüs fonksiyonunun tersi için önce birebirlik ve örtenlik koşulları incelenmelidir.","Reel sayılardan reel sayılara tanımlanan sinüs fonksiyonu birebir değildir çünkü yatay doğru testi birden fazla noktada kesiyor.","Sinüs fonksiyonu örten değildir çünkü değer kümesi -1 ile 1 aralığında olurken, tanım kümesi tüm reel sayılar değildir."]},"beginTime":530,"endTime":673,"href":"/video/preview/12034739842437902857?parent-reqid=1769720205505103-17304330184308580692-balancer-l7leveler-kubr-yp-klg-320-BAL&text=Trigonometric&t=530&ask_summarization=1"},{"index":8,"title":"Sinüs Fonksiyonunun Tersinin Tanım Aralığı","list":{"type":"unordered","items":["Sinüs fonksiyonunun tersini alabilmek için tanım kümesinin elemanları sınırlı olmalıdır.","Sinüs fonksiyonunun tersi için tanım kümesi -π/2 ile π/2 aralığına, değer kümesi ise -1 ile 1 aralığına alınmalıdır.","Bu şekilde sinüs fonksiyonunun tersi vardır, ancak tüm reel sayılardan reel sayılara tanımlı olduğunda tersi yoktur."]},"beginTime":673,"endTime":804,"href":"/video/preview/12034739842437902857?parent-reqid=1769720205505103-17304330184308580692-balancer-l7leveler-kubr-yp-klg-320-BAL&text=Trigonometric&t=673&ask_summarization=1"},{"index":9,"title":"Fonksiyonun Tersinin Alınması","list":{"type":"unordered","items":["Bir fonksiyonun tersi alınırken y gördüğümüz yere x, x gördüğümüz yere y yazılır veya x yalnız bırakılır.","Örneğin, f(x) = 2x + 4 fonksiyonunun tersi f⁻¹(x) = (x - 4)/2 olarak bulunur.","Trigonometrik fonksiyonların tersini almak için önce tanım ve değer kümelerinin aralıkları belirlenmelidir."]},"beginTime":804,"endTime":894,"href":"/video/preview/12034739842437902857?parent-reqid=1769720205505103-17304330184308580692-balancer-l7leveler-kubr-yp-klg-320-BAL&text=Trigonometric&t=804&ask_summarization=1"},{"index":10,"title":"Ters Fonksiyon Kavramı","list":{"type":"unordered","items":["Trigonometrik fonksiyonların tersini almak için \"ark\" kavramı tanımlanır ve sinüs fonksiyonunun tersi arksin olarak gösterilir.","Bir fonksiyonun kendisiyle tersi işleme girdiğinde birbirini yok eder, bu nedenle x = arksin y şeklinde x'i yalnız bırakabiliriz.","Ters trigonometrik fonksiyonlarda içler yer değiştirildiğinde fonksiyonun tersi alınır ve trigonometrik fonksiyonlarda ters işareti \"ark\" olarak gösterilir."]},"beginTime":899,"endTime":1049,"href":"/video/preview/12034739842437902857?parent-reqid=1769720205505103-17304330184308580692-balancer-l7leveler-kubr-yp-klg-320-BAL&text=Trigonometric&t=899&ask_summarization=1"},{"index":11,"title":"Arksin Fonksiyonunun Özellikleri","list":{"type":"unordered","items":["Sinüs 30 derece = 1/2 olduğunda, arksin 1/2 = π/6 olarak hesaplanır ve ters trigonometrik fonksiyonlarda içeride sayı, dışarıda radyan cinsi karşımıza gelir.","Sinüs fonksiyonunun tersi arksin x olarak gösterilir ve tanım aralığı -π/2 ile π/2 arasındadır.","Sinüs fonksiyonunun tersi sadece birinci ve dördüncü bölgelerde tanımlıdır, diğer bölgelerde tanımlı değildir."]},"beginTime":1049,"endTime":1234,"href":"/video/preview/12034739842437902857?parent-reqid=1769720205505103-17304330184308580692-balancer-l7leveler-kubr-yp-klg-320-BAL&text=Trigonometric&t=1049&ask_summarization=1"},{"index":12,"title":"Ters Fonksiyon Örnekleri","list":{"type":"unordered","items":["Arksin 5x-2/3 fonksiyonunun tanım kümesi, 5x-2/3 ifadesinin -1 ile 1 aralığında olması koşuluyla bulunur.","Eşitsizlik çözülerek tanım kümesi -1/5 ile 1 aralığına denk gelir.","f(x) = 2 - 3sinx fonksiyonunun tersi f⁻¹(x) = arksin (2-x)/3 şeklinde yazılır."]},"beginTime":1234,"endTime":1390,"href":"/video/preview/12034739842437902857?parent-reqid=1769720205505103-17304330184308580692-balancer-l7leveler-kubr-yp-klg-320-BAL&text=Trigonometric&t=1234&ask_summarization=1"},{"index":13,"title":"Fonksiyonların Tersi","list":{"type":"unordered","items":["Bir fonksiyonun tersini almak için önce y'yi yalnız bırakmak gerekir.","Bir fonksiyonun tersinin tersi yine kendisidir.","Fonksiyonun tersini alırken içler yer değiştirilir."]},"beginTime":1393,"endTime":1511,"href":"/video/preview/12034739842437902857?parent-reqid=1769720205505103-17304330184308580692-balancer-l7leveler-kubr-yp-klg-320-BAL&text=Trigonometric&t=1393&ask_summarization=1"},{"index":14,"title":"Arksin Fonksiyonu","list":{"type":"unordered","items":["Arksin fonksiyonu, sinüs fonksiyonunun tersidir ve sinüs pozitif olduğunda birinci bölgede, negatif olduğunda dördüncü bölgede incelenir.","Arksin(0) = 0° (0° radyan)","Arksin(1/2) = π/6° (π/6 radyan)","Arksin(1) = π/2° (π/2 radyan)","Arksin(-√3/2) = -π/3° (-π/3 radyan)"]},"beginTime":1511,"endTime":1669,"href":"/video/preview/12034739842437902857?parent-reqid=1769720205505103-17304330184308580692-balancer-l7leveler-kubr-yp-klg-320-BAL&text=Trigonometric&t=1511&ask_summarization=1"},{"index":15,"title":"Fonksiyonların Özellikleri","list":{"type":"unordered","items":["Bir fonksiyonun kendisiyle tersi işleme girdiğinde birbirini götürür.","Sinüs ve arksin fonksiyonları birbirini götürür.","Kosinüs ve arksin fonksiyonları birbirini götürür."]},"beginTime":1669,"endTime":1785,"href":"/video/preview/12034739842437902857?parent-reqid=1769720205505103-17304330184308580692-balancer-l7leveler-kubr-yp-klg-320-BAL&text=Trigonometric&t=1669&ask_summarization=1"},{"index":16,"title":"Trigonometrik İşlemler","list":{"type":"unordered","items":["Tanjant, karşı kenar bölü komşu kenardır.","Kosinüs ve sinüs fonksiyonlarının toplamı veya farkı hesaplanırken, radyan cinsini bilmek önemlidir.","Kosinüs fonksiyonunun dördüncü bölgede pozitif olduğu bilgisi kullanılır."]},"beginTime":1785,"endTime":1958,"href":"/video/preview/12034739842437902857?parent-reqid=1769720205505103-17304330184308580692-balancer-l7leveler-kubr-yp-klg-320-BAL&text=Trigonometric&t=1785&ask_summarization=1"},{"index":17,"title":"Kosinüs Fonksiyonunun Tersi","list":{"type":"unordered","items":["Kosinüs fonksiyonu eksi bir ile bir aralığında değer alır ve pi bölü iki ile pi radyan aralığında birebir ve örten olur.","Kosinüs fonksiyonunun tersini tanımlamak için tanım aralığı pi radyan, değer aralığı ise eksi bir ile bir aralığına alınmalıdır.","Kosinüs fonksiyonunun tersi arc cos x olarak tanımlanır ve birinci ve ikinci bölgede tanımlıdır."]},"beginTime":1962,"endTime":2142,"href":"/video/preview/12034739842437902857?parent-reqid=1769720205505103-17304330184308580692-balancer-l7leveler-kubr-yp-klg-320-BAL&text=Trigonometric&t=1962&ask_summarization=1"},{"index":18,"title":"Ters Fonksiyon Problemi","list":{"type":"unordered","items":["a = 2 arc cos (x+2) fonksiyonunun alabileceği en küçük değer 5/3, en büyük değer ise 2π + 5/3'tür.","a'nın en küçük ve en büyük değerlerinin toplamı 2 + 10/3 olarak 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fonksiyonlar çözüldüğünde, içlerini yer değiştirerek ters fonksiyonun tersi alınır."]},"beginTime":2450,"endTime":2590,"href":"/video/preview/12034739842437902857?parent-reqid=1769720205505103-17304330184308580692-balancer-l7leveler-kubr-yp-klg-320-BAL&text=Trigonometric&t=2450&ask_summarization=1"},{"index":21,"title":"Trigonometrik Denklemler","list":{"type":"unordered","items":["Kotanjant pi bölü iki artı a ifadesi, birim çemberde ikinci bölgeye düşer ve kotanjant ikinci bölgede negatif olduğundan eksi a'ya eşittir.","Tanjant pi bölü iki artı aksin ifadesi, birim çemberde ikinci bölgeye düşer ve tanjant ikinci bölgede negatif olduğundan eksi kotaya eşittir.","Kotanjant pi artı arc kosinüs eksi üç bölü beş ifadesi, birim çemberde üçüncü bölgeye düşer ve kotanjant üçüncü bölgede pozitif olduğundan eksi üç bölü dört olarak bulunur."]},"beginTime":2590,"endTime":2851,"href":"/video/preview/12034739842437902857?parent-reqid=1769720205505103-17304330184308580692-balancer-l7leveler-kubr-yp-klg-320-BAL&text=Trigonometric&t=2590&ask_summarization=1"},{"index":22,"title":"Trigonometrik Eşitsizlikler","list":{"type":"unordered","items":["Arc kosinüs üçte ikisi ifadesi, pozitif olduğu için birinci bölgede tanım aralığına sahiptir.","Tanjant yedi pi eksi x ifadesi, ikinci bölgede tanjant negatif olduğundan eksi tanjant x'e eşittir.","Kotanjantın içine arc sinüs x bölü iki ifadesi, kotanjantın kök üç değerini verdiği için pi bölü altı radyan olarak bulunur."]},"beginTime":2851,"endTime":3161,"href":"/video/preview/12034739842437902857?parent-reqid=1769720205505103-17304330184308580692-balancer-l7leveler-kubr-yp-klg-320-BAL&text=Trigonometric&t=2851&ask_summarization=1"},{"index":23,"title":"Dersin Sonu ve Öneriler","list":{"type":"unordered","items":["Ters trigonometrik fonksiyonlar çözüldüğünde, içlerini yer 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