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Contoh Soal Trigonometri Kelas 10 dan Pembahasannya

Pada kesempatan kali ini, artikel ini akan membahas materi tentang trigonometri kelas 10. Selain materi ada juga soal berikut pembahasan yang berguna bagi meningkatkan pemahaman teman-teman sekalian. Untuk itu langsung saja kita sudah rangkum materi berikut contoh soalnya. Semoga bermanfaat.

Perbandingan Trigonometri Pada Segitiga Siku-Siku

Perhatikan gambar segitiga siku-siku di bawah ini !
trigonometri kelas 10
Berikut perbandingan trigonometri untuk masing-masing lokasi sudut (\(\alpha\hspace{2mm}dan\hspace{2mm}\beta\)) :
\(Sin\hspace{1mm}\alpha=\frac{y}{r}\) \(Sin\hspace{1mm}\beta=\frac{x}{r}\)
\(Cos\hspace{1mm}\alpha=\frac{x}{r}\) \(Cos\hspace{1mm}\beta=\frac{y}{r}\)
\(Tan\hspace{1mm}\alpha=\frac{y}{x}\) \(Tan\hspace{1mm}\beta=\frac{x}{y}\)
\(Cotan\hspace{1mm}\alpha=\frac{x}{y}\) \(Cotan\hspace{1mm}\beta=\frac{y}{x}\)
\(Sec\hspace{1mm}\alpha=\frac{r}{x}\) \(Sec\hspace{1mm}\beta=\frac{r}{y}\)
\(Cosec\hspace{1mm}\alpha=\frac{r}{y}\) \(Cosec\hspace{1mm}\beta=\frac{r}{x}\)



Contoh Soal !


  1. Perhatikan gambar di bawah ini !

    Berdasarkan gambar di atas, tentukanlah :
    1. \(sin\hspace{1mm}\delta\)
    2. \(cos\hspace{1mm}\delta-sin\hspace{1mm}\delta\)
    3. \(tan\hspace{1mm}\gamma+sec\hspace{1mm}\delta\) 
    4. \(cosec\hspace{1mm}\delta+cos\hspace{1mm}\gamma\) 
    5. \(sin\hspace{1mm}(\delta+\gamma)\) 
    6. \(cos\hspace{1mm}(\gamma-\delta)\)
  2. Jika \(sin\hspace{1mm}\alpha=p\), Tentukan nilai dari :
    1. \(cos\hspace{1mm}\alpha+tan\hspace{1mm}\alpha\)
    2. \(tan\hspace{1mm}\alpha\cdot sin\hspace{1mm}\alpha\)

Jawab :

  1. sisi tegak vertikal : \(\sqrt{17^{2}-8^{2}}=\sqrt{289-64}=\sqrt{225}=15\)
    1. \(sin\hspace{1mm}\delta=\frac{depan}{miring}=\frac{15}{17}\)  

    2. \(\begin{eqnarray}cos\hspace{1mm}\delta-sin\hspace{1mm}\delta&=&\frac{8}{17}-\frac{15}{17}\\&=&-\frac{7}{17}\end{eqnarray}\)

    3. \(\begin{eqnarray}tan\hspace{1mm}\gamma+sec\hspace{1mm}\delta&=&\frac{8}{15}+\frac{17}{8}\\&=&\frac{64+255}{120}\\&=&\frac{319}{120}\end{eqnarray}\)

    4. \(\begin{eqnarray}cosec\hspace{1mm}\delta+cos\hspace{1mm}\gamma&=&\frac{17}{15}+\frac{15}{17}\\&=&\frac{289+225}{255}\\&=&\frac{514}{255}\end{eqnarray}\)

    5. \(\begin{eqnarray}sin\hspace{1mm}(\delta+\gamma)&=&sin\hspace{1mm}\delta\cdot cos\hspace{1mm}\gamma+cos\hspace{1mm}\delta\cdot sin\hspace{1mm}\gamma\\&=&\frac{15}{17}\cdot\frac{15}{17}+\frac{8}{17}\cdot\frac{8}{17}\\&=&\frac{225}{289}+\frac{64}{289}\\&=&\frac{289}{289}\\&=&1\end{eqnarray}\)

    6. \(\begin{eqnarray}cos\hspace{1mm}(\gamma-\delta)&=&cos\hspace{1mm}\gamma\cdot cos\hspace{1mm}\delta+sin\hspace{1mm}\gamma\cdot sin\hspace{1mm}\delta\\&=&\frac{15}{17}\cdot\frac{8}{17}+\frac{8}{17}\cdot\frac{15}{17}\\&=&\frac{120}{289}+\frac{120}{289}\\&=&\frac{240}{289}\end{eqnarray}\)
  2. \(sin\hspace{1mm}\alpha=\frac{y}{r}=\frac{p}{1}\)
    maka nilai x :
    \(x=\sqrt{r^{2}-y^{2}}=\sqrt{1^{2}-p^{2}}=\sqrt{1-p^{2}}\) 
    1. \(cos\hspace{1mm}\alpha+tan\hspace{1mm}\alpha=\frac{\sqrt{1-p^{2}}}{1}+\frac{p}{\sqrt{1-p^{2}}}=\frac{1-p^{2}+p}{\sqrt{1-p^{2}}}\)
    2. \(tan\hspace{1mm}\alpha\cdot sin\hspace{1mm}\alpha=\frac{p}{\sqrt{1-p^{2}}}\cdot\frac{p}{1}=\frac{p^{2}}{1-p^{2}}\)

Nilai Trigonometri


Berikut nilai trigonometri untuk masing-masing sudut.



\(0^{o}\)
\(30^{o}\)
\(45^{o}\)
\(60^{o}\)
\(90^{o}\)
Sin
0
\(\frac{1}{2}\)
\(\frac{1}{2}\sqrt{2}\)
\(\frac{1}{2}\sqrt{3}\)
1
Cos
1
\(\frac{1}{2}\sqrt{3}\)
\(\frac{1}{2}\sqrt{2}\)
\(\frac{1}{2}\)
0
Tan
0
\(\frac{1}{3}\sqrt{3}\)
1
\(\sqrt{3}\)
-
Cotan
-
\(\sqrt{3}\)
1
\(\frac{1}{3}\sqrt{3}\)
0
Sec
1
\(\frac{2}{3}\sqrt{3}\)
\(\sqrt{2}\)
2
-
Cosec
-
2
\(\sqrt{2}\)
\(\frac{2}{3}\sqrt{3}\)
1


Kuadran

trigonometri kelas 10
Kuadran I : \(0^{o}<\alpha<90^{o}\)
Kuadran II : \(90^{o}<\alpha<180^{o}\)

Kuadran III : \(180^{o}<\alpha<270^{o}\)
Kuadran IV : \(270^{o}<\alpha<360^{o}\)


Sudut Berelasi


Kuadran I (Semua positif)

\(Cos\hspace{1mm}\alpha=Sin\hspace{1mm}(90^{o}-\alpha)\)
\(Tan\hspace{1mm}\alpha=Cotan\hspace{1mm}(90^{o}-\alpha)\)
\(Cotan\hspace{1mm}\alpha=Tan\hspace{1mm}(90^{o}-\alpha)\)
\(Sec\hspace{1mm}\alpha=Cosec\hspace{1mm}(90^{o}-\alpha)\)
\(Cosec\hspace{1mm}\alpha=Sec\hspace{1mm}(90^{o}-\alpha)\)

Kuadran II (Hanya \(sin\) dan \(cosec\) positif, sisanya negatif)

\(Cos\hspace{1mm}\alpha=-Cos\hspace{1mm}(180^{o}-\alpha)\)
\(Tan\hspace{1mm}\alpha=-Tan\hspace{1mm}(180^{o}-\alpha)\)
\(Cotan\hspace{1mm}\alpha=-Cotan\hspace{1mm}(180^{o}-\alpha)\)
\(Sec\hspace{1mm}\alpha=-Sec\hspace{1mm}(180^{o}-\alpha)\)
\(Cosec\hspace{1mm}\alpha=Cosec\hspace{1mm}(180^{o}-\alpha)\)

Kuadran III (Hanya \(tan\) dan \(cotan\) positif, sisanya negatif)

\(Cos\hspace{1mm}\alpha=-Cos\hspace{1mm}(180^{o}+\alpha)\)
\(Tan\hspace{1mm}\alpha=Tan\hspace{1mm}(180^{o}+\alpha)\)
\(Cotan\hspace{1mm}\alpha=Cotan\hspace{1mm}(180^{o}+\alpha)\)
\(Sec\hspace{1mm}\alpha=-Sec\hspace{1mm}(180^{o}+\alpha)\)
\(Cosec\hspace{1mm}\alpha=-Cosec\hspace{1mm}(180^{o}+\alpha)\)

Kuadran IV (Hanya \(cos\) dan \(sec\) positif, sisanya negatif)

\(Cos\hspace{1mm}\alpha=Cos\hspace{1mm}(360^{o}-\alpha)\)
\(Tan\hspace{1mm}\alpha=-Tan\hspace{1mm}(360^{o}-\alpha)\)
\(Cotan\hspace{1mm}\alpha=-Cotan\hspace{1mm}(360^{o}-\alpha)\)
\(Sec\hspace{1mm}\alpha=Sec\hspace{1mm}(360^{o}-\alpha)\)
\(Cosec\hspace{1mm}\alpha=-Cosec\hspace{1mm}(360^{o}-\alpha)\)


Contoh Soal !

  1. Tentukan nilai dari masing-masing trigonometri di bawah ini :
    1. \(sin\hspace{1mm}120^{o}\)  
    2. \(cos\hspace{1mm}150^{o}\)
    3. \(tan\hspace{1mm}135^{o}\)
    4. \(cosec\hspace{1mm}210^{o}\)
    5. \(sec\hspace{1mm}300^{o}\)
    6. \(sin\hspace{1mm}240^{o}\)
    7. \(cotan\hspace{1mm}315^{o}\)
    8. \(sin\hspace{1mm}420^{o}\)
    9. \(tan\hspace{1mm}495^{o}\)
    10. \(cotan\hspace{1mm}-30^{o}\)
    11. \(sec\hspace{1mm}-120^{o}\)
  2. Jika nilai dari \(sin\hspace{1mm}25^{o}=\frac{a-1}{a+1}\), maka tentukanlah nilai dari :
    1. \(cos\hspace{1mm}65^{o}\)
    2. \(tan\hspace{1mm}155^{o}\)
    3. \(cosec\hspace{1mm}205^{o}\)

Jawab :


    1. \(sin\hspace{1mm}120^{o}=sin\hspace{1mm}(180^{o}-60^{o})=sin\hspace{1mm}60^{o}=\frac{1}{2}\sqrt{3}\)  
    2. \(cos\hspace{1mm}150^{o}=cos\hspace{1mm}(180^{o}-30^{o})=-cos\hspace{1mm}30^{o}=-\frac{1}{2}\sqrt{3}\)  
    3. \(tan\hspace{1mm}135^{o}=tan\hspace{1mm}(180^{o}-45^{o})=-tan\hspace{1mm}45^{o}=-1\)  
    4. \(cosec\hspace{1mm}210^{o}=cosec\hspace{1mm}(180^{o}+30^{o})=-cosec\hspace{1mm}30^{o}=-2\)  
    5. \(sec\hspace{1mm}300^{o}=sec\hspace{1mm}(360^{o}-60^{o})=sec\hspace{1mm}60^{o}=2\)  
    6. \(sin\hspace{1mm}240^{o}=sin\hspace{1mm}(180^{o}+60^{o})=-sin\hspace{1mm}60^{o}=-\frac{1}{2}\sqrt{3}\)  
    7. \(cotan\hspace{1mm}315^{o}=cotan\hspace{1mm}(360^{o}-45^{o})=-cotan\hspace{1mm}{45^{o}}=-1\)  
    8. \(sin\hspace{1mm}420^{o}=sin\hspace{1mm}(360^{o}+60^{o})=sin\hspace{1mm}60^{o}=\frac{1}{2}\sqrt{3}\)  
    9. \(tan\hspace{1mm}495^{o}=tan\hspace{1mm}(360^{o}+135^{o})=tan\hspace{1mm}135^{o}=tan\hspace{1mm}(180^{o}-45^{o})=-tan\hspace{1mm}45^{o}=-1\)
    10. \(cotan\hspace{1mm}(-30^{o})=-cotan\hspace{1mm}30^{o}=-\sqrt{3}\)  
    11. \(sec\hspace{1mm}(-120^{o})=sec\hspace{1mm}120^{o}=sec\hspace{1mm}(180^{o}-60^{o})=sec\hspace{1mm}60^{o}=\frac{1}{3}\sqrt{3}\)
  1. \(sin\hspace{1mm}25^{o}=\frac{y}{r}=\frac{a-1}{a+1}\)
    \(x=\sqrt{r^{2}-y^{2}}=\sqrt{(a+1)^{2}-(a-1)^{2}}=\sqrt{a^{2}+2a+1-a^{2}+2a-1}=\sqrt{4a}=2\sqrt{a}\)
    1. \(cos\hspace{1mm}65^{o}=sin\hspace{1mm}(90^{o}-25^{o})=sin\hspace{1mm}25^{o}=\frac{a-1}{a+1}\)
    2. \(tan\hspace{1mm}155^{o}=tan\hspace{1mm}(180^{o}-25^{o})=-tan\hspace{1mm}25^{o}=-\frac{a-1}{2\sqrt{a}}\)
    3. \(cosec\hspace{1mm}205^{o}=cosec\hspace{1mm}(180^{o}+25^{o})=-cosec\hspace{1mm}25^{o}=-\frac{a+1}{a-1}\)

Identitas Trigonometri

  • \(sin^{2}\theta+cos^{2}\theta=1\)
  • \(1+tan^{2}\theta=sec^{2}\theta\)
  • \(1+cotan^{2}\theta=cosec^{2}\theta\) 
  • \(sec\hspace{1mm}\theta=\frac{1}{cos\hspace{1mm}\theta}\)
  • \(cosec\hspace{1mm}\theta=\frac{1}{sin\hspace{1mm}\theta}\)
  • \(tan\hspace{1mm}\theta=\frac{sin\hspace{1mm}\theta}{cos\hspace{1mm}\theta}=\frac{1}{cotan\hspace{1mm}\theta}\)
  • \(cotan\hspace{1mm}\theta=\frac{cos\hspace{1mm}\theta}{sin\hspace{1mm}\theta}=\frac{1}{tan\hspace{1mm}\theta}\)

Contoh Soal !

  1. \(sec\hspace{1mm}\theta-tan\hspace{1mm}\theta\hspace{1mm}sin\hspace{1mm}\theta=cos\hspace{1mm}\theta\)
  2. \(\frac{cot\hspace{1mm}\theta+tan\hspace{1mm}\theta}{sin\hspace{1mm}\theta\hspace{1mm}cos\hspace{1mm}\theta}=cosec^{2}\hspace{1mm}\theta\hspace{1mm}sec^{2}\hspace{1mm}\theta\)
  3. \(\frac{tan^{2}\hspace{1mm}\theta}{sec\hspace{1mm}\theta}=sec\hspace{1mm}\theta-cos\hspace{1mm}\theta\)
  4. \(\frac{tan\hspace{1mm}\theta}{sec\hspace{1mm}\theta-1}=\frac{sec\hspace{1mm}\theta+1}{tan\hspace{1mm}\theta}\)
  5. \(cot^{2}\hspace{1mm}\theta-cos^{2}\hspace{1mm}\theta=cos^{2}\hspace{1mm}cot^{2}\hspace{1mm}\theta\)

Jawab :


  1. \(\begin{eqnarray}sec\hspace{1mm}\theta-tan\hspace{1mm}\theta\hspace{1mm}sin\hspace{1mm}\theta&=&\frac{1}{cos\hspace{1mm}\theta}-\frac{sin\hspace{1mm}\theta}{cos\hspace{1mm}\theta}\hspace{1mm}sin\hspace{1mm}\theta\\&=&\frac{1}{cos\hspace{1mm}\theta}-\frac{sin^{2}\theta}{cos\hspace{1mm}\theta}\\&=&\frac{1-sin^{2}\theta}{cos\hspace{1mm}\theta}\\&=&\frac{cos^{2}\theta}{cos\hspace{1mm}\theta}\\&=&cos\hspace{1mm}\theta\hspace{5mm}(terbukti)\end{eqnarray}\) 

  2. \(\begin{eqnarray}\frac{cot\hspace{1mm}\theta+tan\hspace{1mm}\theta}{sin\hspace{1mm}\theta\hspace{1mm}cos\hspace{1mm}\theta}&=&\frac{\frac{cos\hspace{1mm}\theta}{sin\hspace{1mm}\theta}+\frac{sin\hspace{1mm}\theta}{cos\hspace{1mm}\theta}}{sin\hspace{1mm}\theta\hspace{1mm}cos\hspace{1mm}\theta}\\&=&\frac{cos^{2}\theta+sin^{2}\theta}{sin\hspace{1mm}\theta\hspace{1mm}cos\hspace{1mm}\theta}\\&=&\frac{\frac{1}{sin\hspace{1mm}\theta\hspace{1mm}cos\hspace{1mm}\theta}}{sin\hspace{1mm}\theta\hspace{1mm}cos\hspace{1mm}\theta}\\&=&\frac{1}{sin^{2}\theta\hspace{1mm}cos^{2}\theta}\\&=&\frac{1}{sin^{2}\theta}\cdot\frac{1}{cos^{2}\theta}\\&=&cosec^{2}\theta\cdot sec^{2}\theta\hspace{5mm}(terbukti)\end{eqnarray}\)

  3. \(\begin{eqnarray}\frac{tan^{2}\hspace{1mm}\theta}{sec\hspace{1mm}\theta}&=&\frac{\frac{sin^{2}\theta}{cos^{2}\theta}}{\frac{1}{cos\hspace{1mm}\theta}}\\&=&\frac{sin^{2}\theta}{cos\hspace{1mm}\theta}\\&=&\frac{1-cos^{2}\theta}{cos\hspace{1mm}\theta}\\&=&\frac{1}{cos\hspace{1mm}\theta}-\frac{cos^{2}\theta}{cos\hspace{1mm}\theta}\\&=&sec\hspace{1mm}\theta-cos\hspace{1mm}\theta\hspace{5mm}(terbukti)\end{eqnarray}\)

  4. \(\begin{eqnarray}\frac{tan\hspace{1mm}\theta}{sec\hspace{1mm}\theta-1}&=&\frac{tan\hspace{1mm}\theta}{\frac{tan^{2}\theta}{sec\hspace{1mm}\theta+1}}\\&=&\frac{sec\hspace{1mm}\theta+1}{tan\hspace{1mm}\theta}\hspace{5mm}(terbukti)\end{eqnarray}\)

  5. \(\begin{eqnarray}cot^{2}\hspace{1mm}\theta-cos^{2}\hspace{1mm}\theta&=&\frac{cos^{2}\theta}{sin^{2}\theta}-cos^{2}\theta\\&=&cos^{2}\theta(\frac{1}{sin^{2}\theta}-1)\\&=&cos^{2}\theta(\frac{1-sin^{2}\theta}{sin^{2}\theta})\\&=&cos^{2}\theta(\frac{cos^{2}\theta}{sin^{2}\theta})\\&=&cos^{2}\theta\cdot cot^{2}\theta\hspace{5mm}(terbukti)\end{eqnarray}\)

Trigonometri Pada Semua Jenis Segitiga (Siku-siku, Sama Kaki, Sama Sisi, Sembarang)

Perhatikan segitiga di bawah ini !

soal trigonometri kelas 10
Keterangan :
a,b, dan c adalah sisi-sisi pada segitiga.
A,B, dan C adalah sudut-sudut pada segitiga.

Aturan Sinus 

\(\frac{a}{sin\hspace{1mm}A}=\frac{b}{sin\hspace{1mm}B}=\frac{c}{sin\hspace{1mm}C}\)


Aturan Cosinus

\(cos\hspace{1mm}A=\frac{b^{2}+c^{2}-a^{2}}{2\cdot b\cdot c}\)
\(cos\hspace{1mm}B=\frac{a^{2}+c^{2}-b^{2}}{2\cdot a\cdot c}\)
\(cos\hspace{1mm}C=\frac{b^{2}+a^{2}-c^{2}}{2\cdot b\cdot a}\)
maka ekivalen dengan :
\(a^{2}=b^{2}+c^{2}-2\cdot b\cdot c\cdot cos\hspace{1mm}A\)

\(b^{2}=a^{2}+c^{2}-2\cdot a\cdot c\cdot cos\hspace{1mm}B\)
\(c^{2}=b^{2}+a^{2}-2\cdot b\cdot a\cdot cos\hspace{1mm}C\)


Luas Segitiga

Jika diketahui dua sisi yang mengapit sebuah sudut, maka kita bisa mencari tahu luas segitiga tersebut.

\(Luas=\frac{1}{2}\cdot a\cdot b\cdot sin\hspace{1mm}C\)
atau
\(Luas=\frac{1}{2}\cdot b\cdot c\cdot sin\hspace{1mm}A\)
atau
\(Luas=\frac{1}{2}\cdot c\cdot a\cdot sin\hspace{1mm}B\)

Contoh Soal !


  1. Perhatikan gambar di bawah ini !
    Tentukanlah :
    1. \(\angle CAB\)
    2. Luas segitiga ABC.
  2. Sebuah segitiga ABC dengan AC = 6 cm, BC = 8 cm, sudut BAC = \(60^{o}\) dan sudut ABC = \(\gamma\). Tentukanlah nilai dari \(sin\hspace{1mm}\gamma\)! 
  3. Diketahui segitiga ABC, dengan AB = 5 cm, BC = 7 cm, dan AC = 8 cm. Sudut BCA = \(\theta\), tentukanlah nilai dari \(tan\hspace{1mm}\theta\) !

Jawab :

  1.  Untuk soal ini kita bisa gunakan aturan sinus.

    1. \(\begin{eqnarray}\frac{CB}{sin\hspace{1mm}\angle CAB}&=&\frac{AC}{sin\hspace{1mm}\angle ABC}\\\frac{10}{sin\hspace{1mm}\angle CAB}&=&\frac{14}{sin\hspace{1mm}63^{o}}\\sin\hspace{1mm}\angle CAB&=&\frac{10\cdot sin\hspace{1mm}63^{o}}{14}\\sin\hspace{1mm}\angle CAB&=&0,636\\\angle CAB&=&sin^{-1}(0,636)\\\angle CAB&=&39,52^{o}\end{eqnarray}\) 
    2. Cari terlebih dahulu sudut C.
      \(\angle C=180^{o}-\angle A-\angle B=180^{o}-39,52^{o}-63^{o}=77,48^{o}\)
      Maka luas segitiga ABC adalah :
      \(L=\frac{1}{2}\cdot 14\cdot 10\cdot sin\hspace{1mm}(77,48^{o})=68,33\) satuan luas.

  2. \(\begin{eqnarray}\frac{6}{sin\hspace{1mm}\gamma}&=&\frac{8}{sin\hspace{1mm}60^{o}}\\sin\hspace{1mm}\gamma&=&\frac{6\cdot sin\hspace{1mm}60^{o}}{8}\\sin\hspace{1mm}\gamma&=&\frac{3}{8}\sqrt{3}\end{eqnarray}\)

  3. \(\begin{eqnarray}cos\hspace{1mm}\theta&=&\frac{8^{2}+7^{2}-5^{2}}{2\cdot 8\cdot 7}\\&=&\frac{88}{112}\\&=&\frac{11}{14}\end{eqnarray}\)
    dengan menggunakan prinsip phytagoras.
    \(cos\hspace{1mm}\theta=\frac{x}{r}=\frac{11}{14}\)
    maka nilai dari : \(y=\sqrt{r^{2}-x^{2}}=\sqrt{14^{2}-11^{2}}=\sqrt{75}=5\sqrt{3}\)
    sehingga nilai dari : \(tan\hspace{1mm}\theta=\frac{y}{x}=\frac{5\sqrt{3}}{11}\)

Itulah materi sekaligus contoh soal trigonometri kelas 10 dan pembahasannya. Terima kasih sudah menyempatkan untuk membaca artikel ini. Untuk itu jika artikel ini bermanfaat bisa kalian share agar bisa bermanfaat juga untuk yang lain. Terima kasih

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