{"id":1803,"date":"2021-05-10T19:25:18","date_gmt":"2021-05-10T19:25:18","guid":{"rendered":"http:\/\/matej.info\/matejmatika\/?page_id=1803"},"modified":"2021-05-30T11:59:55","modified_gmt":"2021-05-30T11:59:55","slug":"trigonometrija-teorija","status":"publish","type":"page","link":"http:\/\/matej.info\/matejmatika\/trigonometrija-teorija\/","title":{"rendered":"Trigonometrija-teorija"},"content":{"rendered":"\n
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TRIGONOMETRI\u010cNE FUNKCIJE<\/h1>\n\n\n\n

<\/p>\n\n\n\n

OSNOVNE ZVEZE MED KOTNIMI FUNKCIJAMI<\/h4><\/path><\/svg><\/div>
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Med kotnimi funkcijami istega kota \"x\" velja: <\/a><\/p>\n\n\n

(1) <\/span>   <\/span>\"\begin{eqnarray*}<\/p>\n

(2) <\/span>   <\/span>\"\begin{eqnarray*}<\/p>\n

(3) <\/span>   <\/span>\"\begin{eqnarray*}<\/p>\n

(4) <\/span>   <\/span>\"\begin{eqnarray*}<\/p>\n

(5) <\/span>   <\/span>\"\begin{eqnarray*}<\/p>\n

(6) <\/span>   <\/span>\"\begin{eqnarray*}<\/p>\n\n\n\n

Primer: Poenostavi izraz:<\/p>\n\n\n

\"(1-\sin^2 x) \cdot \cos^-^2 x =\"<\/p>\n

Najprej izra\u010dunamo \"(1-\sin^2 x)\". Ker poznamo osnovno zvezo \"\sin^2 x + \cos^2 x &=& 1\", dobimo rezultat \"\cos^2 x\". \"\cos^-^2 x\", pa postavimo v ulomek, saj negativna potenca pomeni obratno vrednost v tem primeru \"\frac{1}. Iz tega sledi, da izra\u010dunamo ena\u010dbo \"\frac{\cos^2 x}{\cos^2 x}\", kar je 1.<\/p>\n<\/div><\/div><\/div><\/div><\/div><\/div>\n\n\n\n

TABELA KOTNIH FUNKCIJ in PREHOD NA OSTRI KOT<\/h4><\/path><\/svg><\/div>
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   <\/span>   <\/span>\"\[\begin{array}{|l|c|c|c|c|c|}<\/p>\n<\/code><\/pre>\n\n\n\n
Pri ra\u010dunanju vrednosti kotnih funkcij upo\u0161tevamo formule za prehod na ostri kot.<\/p>\n\n\n
 (7) <\/span>   <\/span>\"\begin{eqnarray*}<\/p>\n\n\n\n
 <\/p>\n<\/div><\/div><\/div><\/div><\/div><\/div>\n\n\n\n
RADIANI<\/h4><\/path><\/svg><\/div>
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En radian(rd) je kot, ki mu pripada kro\u017eni lok z dol\u017eino enega polmera kro\u017enice.\n\"\pi = 180^{\circ}\"\n\"l= \frac{\pi r \alpha}{180^{\circ}}\"\n\"\alpha= \frac{180^{\circ}}{\pi}\" \n\"1^{\circ} = \frac{\pi}{180}\"\nZapi\u0161i kote v radianih:\n\"0^{\circ} = 0\"\n\"30^{\circ} = \frac{180}{6} = \frac{\pi}{6}\"\n\"29^{\circ} = 29\cdot 1^{\circ} = 29\cdot \frac{\pi}{180}=\frac{29\pi}{180}= 0,51\"\n\"45^{\circ} = \frac{180}{4} = \frac{\pi}{4}\"\n\"60^{\circ} = \frac{180}{3} = \frac{\pi}{3}\"\n\"210^{\circ} = 180^{\circ}+30^{\circ} = \pi + \frac{\pi}{6}= \frac{7\pi}{6}\"\n\"315^{\circ} = 360^{\circ}-45^{\circ} = 2\pi - \frac{\pi}{4}= \frac{7\pi}{4}\"\n\"270^{\circ} = 180^{\circ}+90^{\circ} = \pi + \frac{\pi}{2}= \frac{3\pi}{2}\"\n\"\frac {9\pi}{2}= 9\cdot 90^{\circ}= 810^{\circ}\"\n\"\frac {4\pi}{3}= 4\cdot 60^{\circ}= 240^{\circ}\"\n<\/code><\/pre>\n\n\n\n
<\/p>\n\n\n\n
 <\/p>\n<\/div><\/div><\/div><\/div><\/div><\/div>\n\n\n\n
ADICIJSKI IZREKI<\/h4><\/path><\/svg><\/div>
 (8) <\/span>   <\/span>\"\begin{eqnarray*}<\/p>\n
 (9) <\/span>   <\/span>\"\begin{eqnarray*}<\/p>\n
 (10) <\/span>   <\/span>\"\begin{eqnarray*}<\/p>\n
 (11) <\/span>   <\/span>\"\begin{eqnarray*}<\/p>\n
 (12) <\/span>   <\/span>\"\begin{eqnarray*}<\/p>\n
 (13) <\/span>   <\/span>\"\begin{eqnarray*}<\/p>\n\n\n\n
Primer<\/span>: Naj bo \"\sin x=\dfrac{7}{25}\" in \"x\" topi kot ter \"\cos y=\dfrac{4}{5}\", kjer je \"-\dfrac{3\pi}{2}<y<\2\pi.\" Izra\u010dunaj vrednosti \"\sin (x+y)\", \"\cos(x+y)\", \"\tan(x+y)\" ter \"\cot(x+y).\"<\/p>\n\n\n\n
Najprej izra\u010dunamo \"\cos^2 x=1-\sin^2 x= 1-(\frac{7}{25})^2=\dfrac{576}{625}.\" Ker je \"x\" topi kot, je \"\cos x=-\dfrac{24}{25}.\" Izra\u010dunamo tudi \"\sin^2 y=1-cos^2 y=1-(\dfrac{4}{5})^2=\dfrac{9}{25}.\" Ker je \"y\" v \u010detrtem kvadrantu , sledi \"\sin y=-\frac{3}{5}\".  Nato uporabimo adicijske izreke: 
   <\/span>   <\/span>\"\[\sin(x+y)=\dfrac{7}{25}\cdot \dfrac{4}{5}+\dfrac{24}{25}\left(-\dfrac{3}{5}\right)=-\frac{44}{125},\]\"<\/p> 
   <\/span>   <\/span>\"\[\cos(x+y)=\dfrac{24}{25}\cdot \dfrac{4}{5}-\dfrac{7}{25}\left(-\dfrac{3}{5}\right)=-\frac{117}{125}.\]\"<\/p><\/p>\n\n\n\n
Izra\u010dunamo \u0161e \"\tan x=\dfrac{\sin x}{\cos x}=-\dfrac{7}{24}\", \"\tan y=\dfrac{\sin y}{\cos y}=-\dfrac{3}{4},\" nato pa \u0161e po adicijskem izreku 
   <\/span>   <\/span>\"\[\tan(x+y)= \dfrac{(-\frac{7}{24})+(-\frac{3}{4})}{1-(-\frac{7}{24})\cdot(-\frac{3}{4})}=-\dfrac{4}{3}.\]\"<\/p> Za \"\cot(x+y)\" uporabimo formulo: 
   <\/span>   <\/span>\"\[\cot(x+y)=\dfrac{1}{\tan(x+y)}=-\dfrac{3}{4}.\]\"<\/p><\/p>\n<\/div><\/div><\/div><\/div><\/div><\/div>\n\n\n\n
FUNKCIJE DVOJNIH IN POLOVI\u010cNIH KOTOV<\/h4><\/path><\/svg><\/div>
 (14) <\/span>   <\/span>\"\begin{eqnarray*}<\/p>\n
 (15) <\/span>   <\/span>\"\begin{eqnarray*}<\/p>\n
 (16) <\/span>   <\/span>\"\begin{eqnarray*}<\/p>\n
 (17) <\/span>   <\/span>\"\begin{eqnarray*}<\/p>\n
 (18) <\/span>   <\/span>\"\begin{eqnarray*}<\/p>\n
 (19) <\/span>   <\/span>\"\begin{eqnarray*}<\/p>\n\n\n\n
Primer: Izra\u010dunaj \"\sin2\alpha\", \"\cos2\alpha\", \"\tan2\alpha\", \u010de je \"\sin\alpha =\frac{8}{17}\". Kot \"\alpha\" je oster kot.<\/p>\n\n\n
Najprej izra\u010dunamo \"cos\alpha\" iz formule \"cos^2\alpha = 1 - \sin^2\alpha\" . V ena\u010dbo ustavimo \"sin\alpha\", ki ga skvadriramo in od\u0161tejemo od 1. Tega nato korenimo in dobimo \"cos\alpha = \frac{15}{17}\". Sedaj lahko izra\u010dunamo prvi primer:
\n\"\sin2\alpha = 2\sin\alpha \cdot \cos\alpha = 2\cdot \frac{8}{17} \cdot \frac{15}{17} =\frac{240}{289}\"<\/p>\n
Za naslednji primer rabimo skvadrirati \u017ee izra\u010dunana primera \"sin\alpha\" in \"cos\alpha\":
\n\"\cos2\alpha = \cos^2 \alpha - \sin^2 \alpha = \frac{225}{289} - \frac{64}{289} =\frac{161}{289}\"<\/p>\n
Za izra\u010dun \"\tan^2 \alpha\", najprej izra\u010dunamo \"\tan\alpha\" s formulo \"\tan\alpha= \frac{\sin\alpha}{cos\alpha}\". Ko v njo vnesemo \u017ee pridobljena \"\sin\alpha\" in \"\cos\alpha\", dobimo, da je \"\tan\alpha=\frac{8}{15}\". Sedaj lahko re\u0161imo dano nalogo:
\n\"\tan^2 \alpha = \frac {2\tan\alpha}{1-tan^2 \alpha} = \frac {2\cdot \frac{8}{15}}{1 - (\frac{8}{15})^2} = \frac{\frac{16}{15}} {\frac{161}{15^2}} = \frac{16\cdot 15}{161} = \frac{240}{161}\"<\/p>\n<\/div><\/div><\/div><\/div><\/div><\/div>\n\n\n\n
FAKTORIZACIJA KOTNIH FUNKCIJ<\/h4><\/path><\/svg><\/div>
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   <\/span>   <\/span>\"\[\sin x + \sin y = 2\sin\frac {x+y}{2} \cdot \cos\frac {x-y}{2}\]\"<\/p><\/p>\n\n\n\n
   <\/span>   <\/span>\"\[\sin x - \sin y = 2\sin\frac {x-y}{2} \cdot \cos\frac {x+y}{2}\]\"<\/p><\/p>\n\n\n\n
   <\/span>   <\/span>\"\[\cos x + \cos y = 2\cos\frac {x+y}{2} \cdot \cos\frac {x-y}{2}\]\"<\/p><\/p>\n\n\n\n
   <\/span>   <\/span>\"\[\cos x - \cos y = -2\cos\frac {x+y}{2} \cdot \cos\frac {x-y}{2}\]\"<\/p><\/p>\n\n\n\n
<\/p>\n<\/div><\/div><\/div><\/div><\/div><\/div>\n\n\n\n
RAZ\u010cLENJEVANJE KOTNIH FUNKCIJ<\/h4><\/path><\/svg><\/div>
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   <\/span>   <\/span>\"\[\sin x \cdot \sin y = -\frac {1}{2} \cdot (\cos(x+y) - \cos(x-y))\]\"<\/p><\/p>\n\n\n\n
   <\/span>   <\/span>\"\[\cos x \cdot \cos y = \frac {1}{2} \cdot (\cos(x+y) + \cos(x-y))\]\"<\/p><\/p>\n\n\n\n
   <\/span>   <\/span>\"\[\sin x \cdot \cos y = \frac {1}{2} \cdot (\sin(x+y) + \sin(x-y))\]\"<\/p><\/p>\n<\/div><\/div><\/div><\/div><\/div><\/div>\n\n\n\n
GRAFI KOTNIH FUNKCIJ<\/h4><\/path><\/svg><\/div>
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  1. \"f(x)=\sin x.\"  Zapi\u0161i lastnosti funkcije <\/li><\/ol>\n\n\n
    \"Rendered<\/p>\n\n\n\n
    2. Kosinus<\/strong>: \"f(x)=\cos x\" Zapi\u0161i lastnosti funkcije in nari\u0161i graf.<\/p>\n\n\n
    \"Rendered<\/p>\n\n\n\n
    3. Tangens<\/strong>: \"f(x)=\tan x.\" Zapi\u0161i lastnosti funkcije in nari\u0161i graf.<\/p>\n\n\n
    \"Rendered<\/p>\n\n\n\n
    4. Kotangens<\/strong>: \"f(x)=\cot x.\" Zapi\u0161i lastnosti funkcije in nari\u0161i graf.<\/p>\n\n\n
    \"Rendered<\/p>\n<\/div><\/div><\/div><\/div><\/div><\/div>\n\n\n\n
    <\/p>\n<\/div>\n<\/div><\/section>\n\n\n\n
    <\/div>
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    \n
    KRO\u017dNE FUNKCIJE<\/h1>\n\n\n\n
    <\/p>\n\n\n\n
    DEFINICIJA KRO\u017dNIH (CIKLOMETRI\u010cNIH) FUNKCIJ<\/h4><\/path><\/svg><\/div>
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    <\/p>\n<\/div><\/div><\/div><\/div><\/div><\/div>\n\n\n\n
    GRAFI KRO\u017dNIH FUNKCIJ<\/h4><\/path><\/svg><\/div>
    \n
    Description for this block. Use this space for describing your block. Any text will do. Description for this block. You can use this space for describing your block. Description for this block. Use this space for describing your block. Any text will do. Description for this block. You can use this space for describing your block.<\/p>\n<\/div><\/div><\/div><\/div><\/div><\/div>\n\n\n\n
    TRIGONOMETRI\u010cNE ENA\u010cBE<\/h4><\/path><\/svg><\/div>
    \n
    1. \"f(x)=\sin x.\"  Zapi\u0161i lastnosti funkcije <\/li><\/ol>\n\n\n
      \"Rendered<\/p>\n\n\n\n
      2. Kosinus<\/strong>: \"f(x)=\cos x\" Zapi\u0161i lastnosti funkcije in nari\u0161i graf.<\/p>\n\n\n
      \"Rendered<\/p>\n\n\n\n
      3. Tangens<\/strong>: \"f(x)=\tan x.\" Zapi\u0161i lastnosti funkcije in nari\u0161i graf.<\/p>\n\n\n
      \"Rendered<\/p>\n\n\n\n
      4. Kotangens<\/strong>: \"f(x)=\cot x.\" Zapi\u0161i lastnosti funkcije in nari\u0161i graf.<\/p>\n\n\n
      \"Rendered<\/p>\n<\/div><\/div><\/div><\/div><\/div><\/div>\n\n\n\n
      UPORABA TRIGONOMETRIJE<\/h4><\/path><\/svg><\/div>
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      1. kotne funkcije v pravokotnem trikotniku<\/li>
      2. kosinusni izrek<\/li>
      3. sinusni izrek<\/li><\/ol>\n<\/div><\/div><\/div><\/div><\/div><\/div>\n\n\n\n
        <\/p>\n<\/div>\n<\/div><\/section>\n\n\n\n
        <\/div>
        \n
        <\/div>\n\n\n\n
        <\/div>\n\n\n\n
        <\/div>\n<\/div><\/section>\n<\/div>\n<\/div><\/section>\n\n\n\n\n\n
        <\/p>\n","protected":false},"excerpt":{"rendered":"","protected":false},"author":1,"featured_media":297,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"template-page-builder.php","meta":{"footnotes":""},"class_list":["post-1803","page","type-page","status-publish","has-post-thumbnail","hentry","grid-sizer"],"featured_image_urls_v2":{"full":["http:\/\/matej.info\/matejmatika\/wp-content\/uploads\/2021\/03\/TR.png",705,508,false],"thumbnail":["http:\/\/matej.info\/matejmatika\/wp-content\/uploads\/2021\/03\/TR-150x150.png",150,150,true],"medium":["http:\/\/matej.info\/matejmatika\/wp-content\/uploads\/2021\/03\/TR-300x216.png",300,216,true],"medium_large":["http:\/\/matej.info\/matejmatika\/wp-content\/uploads\/2021\/03\/TR.png",705,508,false],"large":["http:\/\/matej.info\/matejmatika\/wp-content\/uploads\/2021\/03\/TR.png",705,508,false],"post-slider-thumb-size":["http:\/\/matej.info\/matejmatika\/wp-content\/uploads\/2021\/03\/TR-330x190.png",330,190,true],"post-category-slider-size":["http:\/\/matej.info\/matejmatika\/wp-content\/uploads\/2021\/03\/TR.png",330,238,false],"1536x1536":["http:\/\/matej.info\/matejmatika\/wp-content\/uploads\/2021\/03\/TR.png",705,508,false],"2048x2048":["http:\/\/matej.info\/matejmatika\/wp-content\/uploads\/2021\/03\/TR.png",705,508,false],"blossom-coach-schema":["http:\/\/matej.info\/matejmatika\/wp-content\/uploads\/2021\/03\/TR-600x60.png",600,60,true],"blossom-coach-slider":["http:\/\/matej.info\/matejmatika\/wp-content\/uploads\/2021\/03\/TR.png",705,508,false],"blossom-coach-fullwidth":["http:\/\/matej.info\/matejmatika\/wp-content\/uploads\/2021\/03\/TR.png",705,508,false],"blossom-coach-with-sidebar":["http:\/\/matej.info\/matejmatika\/wp-content\/uploads\/2021\/03\/TR-705x500.png",705,500,true],"blossom-coach-latest":["http:\/\/matej.info\/matejmatika\/wp-content\/uploads\/2021\/03\/TR-540x400.png",540,400,true]},"post_excerpt_stackable_v2":"
        TRIGONOMETRI\u010cNE FUNKCIJE OSNOVNE ZVEZE MED KOTNIMI FUNKCIJAMI Med kotnimi funkcijami istega kota velja: (1)   (2)   (3)   (4)   (5)   (6)   Primer: Poenostavi izraz: Najprej izra\u010dunamo . Ker poznamo osnovno zvezo , dobimo rezultat . , pa postavimo v ulomek, saj negativna potenca pomeni obratno vrednost v tem primeru . Iz tega sledi, da izra\u010dunamo ena\u010dbo , kar je 1. TABELA KOTNIH FUNKCIJ in PREHOD NA OSTRI KOT     Pri ra\u010dunanju vrednosti kotnih funkcij upo\u0161tevamo formule za prehod na ostri kot. (7)   RADIANI En radian(rd) je kot, ki mu pripada kro\u017eni lok z dol\u017eino…<\/p>\n","category_list_v2":"","author_info_v2":{"name":"profM","url":"http:\/\/matej.info\/matejmatika\/author\/profm\/"},"comments_num_v2":"0 comments","_links":{"self":[{"href":"http:\/\/matej.info\/matejmatika\/wp-json\/wp\/v2\/pages\/1803","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/matej.info\/matejmatika\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"http:\/\/matej.info\/matejmatika\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"http:\/\/matej.info\/matejmatika\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/matej.info\/matejmatika\/wp-json\/wp\/v2\/comments?post=1803"}],"version-history":[{"count":327,"href":"http:\/\/matej.info\/matejmatika\/wp-json\/wp\/v2\/pages\/1803\/revisions"}],"predecessor-version":[{"id":2384,"href":"http:\/\/matej.info\/matejmatika\/wp-json\/wp\/v2\/pages\/1803\/revisions\/2384"}],"wp:featuredmedia":[{"embeddable":true,"href":"http:\/\/matej.info\/matejmatika\/wp-json\/wp\/v2\/media\/297"}],"wp:attachment":[{"href":"http:\/\/matej.info\/matejmatika\/wp-json\/wp\/v2\/media?parent=1803"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}