Jika segitiga ABC dengan sisi miring = 9cm, dan sisi depan = 3cm, maka tentukanlah nilai dari:
A.) sin A
B.) cos A
C.) tan A
D.) csc A
E.) cot A
[tex]\:[/tex]
[tex]\hookrightarrow \boxed{\sf{1soal = 20poin.}}[/tex]
Jawaban:
A.) sin A = ⅓
B.) cos A = ⅔√2
C.) tan A = ¼√2
D.) csc A = 3
E.) cot A = 2√2
Penjelasan dengan langkah-langkah:
[tex] \mathbb{ \underline {TRIGONOMETRI}}[/tex]
diketahui
- depan = 3
- miring = 9
cari sisi samping
[tex] \rm samping {}^{2} = {9}^{2} - {3}^{2} [/tex]
[tex] \rm samping {}^{2} = 81 - 9 = 72[/tex]
[tex] \rm samping = \sqrt{72} = 6 \sqrt{2} [/tex]
...
penyelesaian soal
[tex]\rm \sin \: A = \frac{de}{mi} = \frac{3}{9 } = \frac{1}{3} [/tex]
[tex]_____________[/tex]
[tex]\rm \cos\: A = \frac{sa}{mi} = \frac{6 \sqrt{2} }{9} [/tex]
[tex]\rm \cos\: A = \frac{2}{3} \sqrt{2} [/tex]
[tex]_____________[/tex]
[tex]\rm \tan\: A = \frac{de}{sa} = \frac{3}{6 \sqrt{2} } [/tex]
[tex]\rm \tan\: A = \frac{3}{6 \sqrt{2} } \times \frac{6 \sqrt{2} }{6 \sqrt{2} } = \frac{18 \sqrt{2} }{36 \times 2} [/tex]
[tex]\rm \tan \: A= \frac{18}{72} \sqrt{2} = \frac{1}{4} \sqrt{2} [/tex]
[tex]_____________[/tex]
[tex]\rm \csc \: A= \frac{1}{ \sin} = \frac{mi}{de} = \frac{9}{3} = 3[/tex]
[tex]_____________[/tex]
[tex]\rm \cot \: A= \frac{1}{ \tan} = \frac{sa}{de} = \frac{6 \sqrt{2} }{3} [/tex]
[tex]\rm \cot\: A = \frac{6}{3} \sqrt{2} = 2 \sqrt{2} [/tex]
Jika segitiga ABC dengan sisi miring = 9cm, dan sisi depan = 3cm, maka :
- Sisi sampingnya 6√2
[tex]\boxed{\bf{A.) \ \sin A=\frac{1}{3}cm}}[/tex]
[tex]\boxed{\bf{B.) \ \cos A=\frac{2\sqrt{2}}{3}cm}}[/tex]
[tex]\boxed{\bf{C.) \ \tan A=\frac{\sqrt{2}}{4}cm}}[/tex]
[tex]\boxed{\bf{D.) \ \csc A=3cm}}[/tex]
[tex]\boxed{\bf{E.) \ \cot A=2\sqrt{2}cm}}[/tex]
[tex] \: [/tex]
Trigonometri
Pendahuluan
A.) Definisi
Perbandingan Trigonometri
Pada segitiga siku-siku ABC, berlaku :
*Gambar ke-1
[tex]\small\mathbf{\left(a.\right)\ \ \sin\alpha=\frac{y}{r}=\frac{de}{mi}} [/tex]
[tex]\small\mathbf{\left(b.\right)\ \ \cos\alpha=\frac{x}{r}=\frac{sa}{mi}} [/tex]
[tex]\small\mathbf{\left(c.\right)\ \ \tan\alpha=\frac{y}{x}=\frac{de}{sa}} [/tex]
[tex]\small\mathbf{\left(d.\right)\ \ \csc\alpha=\frac{1}{\sin\alpha}=\frac{r}{y}}[/tex]
[tex]\small\mathbf{\left(e.\right)\ \ \sec\alpha=\frac{1}{\cos\alpha}=\frac{r}{x}}[/tex]
[tex]\small\mathbf{\left(f.\right)\ \ \cot\alpha=\frac{1}{\tan\alpha}=\frac{y}{x}}[/tex]
[tex] \: [/tex]
B.) Sudut dan Kuadran
1.) Pembagian Daerah
[tex]\boxed{\begin{array}{c|c|c|c|c}\underline{\mathbf{Kuadran}}&\underline{\mathbf{I}}&\underline{\mathbf{II}}&\underline{\mathbf{III}}&\underline{\mathbf{IV}}\\&&&\\\mathbf{absis(x)}&\mathbf{+}&\mathbf{-}&\mathbf{-}&\mathbf{+}\\&&&\\\mathbf{Ordinat(y)}&\mathbf{+}&\mathbf{+}&\mathbf{-}&\mathbf{-}\end{array}}[/tex]
2.) Tanda-tanda Fungsi
[tex]\boxed{\begin{array}{c|c|c|c|c}\underline{\mathbf{Kuadran}}&\underline{\mathbf{I}}&\underline{\mathbf{II}}&\underline{\mathbf{III}}&\underline{\mathbf{IV}}\\&&&\\\mathbf{sin}&\mathbf{+}&\mathbf{+}&\mathbf{-}&\mathbf{-}\\&&&\\\mathbf{cos}&\mathbf{+}&\mathbf{-}&\mathbf{-}&\mathbf{+}\\&&&\\\mathbf{tan}&\mathbf{+}&\mathbf{-}&\mathbf{+}&\mathbf{-}\end{array}}[/tex]
3.) Sudut-sudut Istimewa
[tex]\boxed{\begin{array}{c|c|c|c|c}\underline{\mathbf{Kuadran}}&\underline{\mathbf{0^{\circ}}}&\underline{\mathbf{30^{\circ}}}&\underline{\mathbf{45^{\circ}}}&\underline{\mathbf{60^{\circ}}}\\&&&\\\mathbf{sin}&\mathbf{0}&\mathbf{\frac{1}{2}}&\mathbf{\frac{1}{2}\sqrt{2}}&\mathbf{\frac{1}{2}\sqrt{3}}\\&&&\\\mathbf{cos}&\mathbf{1}&\mathbf{\frac{1}{2}\sqrt{3}}&\mathbf{\frac{1}{2}\sqrt{2}}&\mathbf{\frac{1}{2}}\\&&&\\\mathbf{tan}&\mathbf{0}&\mathbf{\frac{1}{3}\sqrt{3}}&\mathbf{1}&\mathbf{\sqrt{3}}\end{array}} [/tex] [tex] \boxed{\begin{array}{c}\underline{\mathbf{90^{\circ}}}\\\\\mathbf{1}\\\\\mathbf{0}\\\\\infty\end{array}} [/tex]
[tex] \: [/tex]
[tex] \: [/tex]
Pembahasan
Diketahui :
Segitiga ABC dengan sisi miring = 9cm, dan sisi depan = 3cm
Ditanya :
tentukanlah nilai dari:
A.) sin A
B.) cos A
C.) tan A
D.) csc A
E.) cot A
Jawaban :
berarti,
[tex]\boxed{\bf{sisi\ miring=9cm \to mi}}[/tex]
[tex]\boxed{\bf{sisi\ depan=3cm \to de}}[/tex]
[tex]\bf{sisi\ samping=\sqrt{9^{2}-3^{2}}}[/tex]
[tex]\bf{sisi\ samping=\sqrt{72}}[/tex]
[tex]\boxed{\bf{sisi\ samping=6\sqrt{2} \to sa}}[/tex]
A.) sin A
[tex]\bf{\sin A=\frac{de}{mi}}[/tex]
[tex]\bf{\sin A=\frac{3}{9}cm}[/tex]
[tex]\boxed{\bf{\sin A=\frac{1}{3}cm}}[/tex]
[tex] \: [/tex]
B.) cos A
[tex]\bf{\cos A=\frac{sa}{mi}}[/tex]
[tex]\bf{\cos A=\frac{6\sqrt{2}}{9}cm}[/tex]
[tex]\boxed{\bf{\cos A=\frac{2\sqrt{2}}{3}cm}}[/tex]
[tex] \: [/tex]
C.) tan A
[tex]\bf{\tan A=\frac{de}{sa}}[/tex]
[tex]\bf{\tan A=\frac{3}{6\sqrt{2}}cm}[/tex]
[tex]\bf{\tan A=\frac{1}{2\sqrt{2}}cm}[/tex]
[tex]\bf{\tan A=\frac{2\sqrt{2}}{8}cm}[/tex]
[tex]\boxed{\bf{\tan A=\frac{\sqrt{2}}{4}cm}}[/tex]
[tex] \: [/tex]
D.) csc A
[tex]\bf{\csc A=\frac{1}{\sin A}}[/tex]
[tex]\bf{\csc A=\frac{1}{\frac{1}{3}}cm}[/tex]
[tex]\boxed{\bf{\csc A=3cm}}[/tex]
[tex] \: [/tex]
E.) cot A
[tex]\bf{\cot A=\frac{1}{\tan A}}[/tex]
[tex]\bf{\cot A=\frac{1}{\frac{\sqrt{2}}{4}}cm}[/tex]
[tex]\bf{\cot A=\frac{4}{\sqrt{2}}cm}[/tex]
[tex]\bf{\cot A=\frac{4\sqrt{2}}{2}cm}[/tex]
[tex]\boxed{\bf{\cot A=2\sqrt{2}cm}}[/tex]
[tex] \: [/tex]
[tex] \: [/tex]
Pelajari Lebih Lanjut :
- Contoh soal mencari sisi samping : https://brainly.co.id/tugas/48680192
- Contoh soal dan penyelesaian trigonometri : https://brainly.co.id/tugas/14823036
- Contoh soal yang serupa 1 : https://brainly.co.id/tugas/9349166
- Contoh soal yang serupa 2 : https://brainly.co.id/tugas/14975792
- Mencari cos a jika diketahui sin a : https://brainly.co.id/tugas/14652547
[tex] \: [/tex]
[tex] \: [/tex]
Detail Jawaban :
Grade : SMA
Kode Kategorisasi : 10.2.6
Kelas : 10
Kode Mapel : 2
Pelajaran : Matematika
Bab : 6
Sub Bab : Bab 6 – Trigonometri Dasar
[tex] \: [/tex]
Kata Kunci : Trigonometri, sisi depan, sisi samping, sisi miring.
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