Identitas Trigonometri

Identitas dasar yang merupakan hubungan kebalikan
Setelah mengetahui dasar-dasar dari trigonometri, maka selanjutnya adalah identitas dasar trigonometri yang merupakan hubungan kebalikan.

• [latex]\large cosec\: \alpha =\frac{1}{sin\: \alpha }[/latex]
• [latex]\large sec\: \alpha =\frac{1}{cos\: \alpha }[/latex]
• [latex]\large cotan\: \alpha =\frac{1}{tan\: \alpha }[/latex]

Identitas dasar yang merupakan hubungan perbandingan

• [latex]\large tan \: \alpha = \frac{sin \: \alpha }{cos \: \alpha }\Rightarrow cotan \: \alpha =\frac{cos \: \alpha }{sin\: \alpha }[/latex]

 Identitas dasar yang merupakan hubungan Pythagoras

• [latex]\large sin^{2}\: \alpha +cos^{2}\: \alpha =1[/latex]

Pembuktian [latex]\large sin^{2}\: \alpha +cos^{2}\: \alpha =1[/latex] dengan menggunakan pythagoras

Berdasarkan teorema Pythagoras diketahui bahwa:

[latex]\large c^{2}=a^{2}+b^{2}[/latex]

[latex]\large sin^{2} \alpha = \frac{a}{c} \rightarrow sin^{2}\alpha = \left ( \frac{b}{c} \right )^{2}\rightarrow sin^{2}\alpha =\frac{a^{2}}{c^{2}}[/latex]

[latex]\large cos^{2} \alpha = \frac{a}{c} \rightarrow cos^{2}\alpha = \left ( \frac{b}{c} \right )^{2}\rightarrow cos^{2}\alpha =\frac{a^{2}}{c^{2}}[/latex]

[latex]\large sin^{2}\alpha +cos^{2}\alpha = \frac{a^{2}}{c^{2}} + \frac{b^{2}}{c^{2}}[/latex]

[latex]\rightarrow \frac{a^{2}+b^{2}}{c^{2}} \rightarrow Ingat \: Bahwa \; c^{2}=a^{2}+b^{2}[/latex]

[latex]\rightarrow \frac{c^{2}}{c^{2}}= 1[/latex]

• [latex]\large sec^{2}\: \alpha – tan^{2}\: \alpha =1[/latex]
• [latex]\large cosec^{2}\: \alpha +cotan^{2}\: \alpha =1[/latex]

Dari identitas trigonometri di atas, maka diperoleh identitas trigonometri lainnya sebagai berikut:
1. [latex]\large sin^{2}\: \alpha +cos^{2}\: \alpha =1[/latex]

• [latex]\large sin^{2}\: \alpha = 1-cos^{2}\: \alpha[/latex]
• [latex]\large cos^{2}\: \alpha = 1-sin^{2}\: \alpha[/latex]

2. [latex]\large sec^{2}\: \alpha +cos^{2}\alpha =1[/latex]

[latex]\large \Rightarrow sin^{2}\: \alpha +cos^{2}\alpha =1[/latex]   (masing-masing ruas dikalikan dengan [latex]\large \frac{1}{cos^{2}\: \alpha }[/latex] )

[latex]\large \Rightarrow \frac{sin^{2}\: \alpha }{cos^{2}\: \alpha }+\frac{sin^{2}\: \alpha }{cos^{2}\: \alpha }=\frac{1}{cos^{2}\: \alpha }[/latex]

[latex]\large \Rightarrow \frac{sin^{2}\: \alpha }{cos^{2}\: \alpha }+1=\frac{1}{cos^{2}\: \alpha }[/latex]

[latex]\large \Rightarrow tan^{2}\: \alpha +1 = sec^{2}\: \alpha[/latex]

[latex]\large \Rightarrow sec^{2}\: \alpha =1 +tan^{2}\: \alpha[/latex]

3. [latex]\large cosec^{2}\: \alpha =1 + cotan^{2}\: \alpha[/latex]

[latex]\large \Rightarrow sec^{2}\: \alpha – tan^{2}\: \alpha = 1[/latex] (masing-masing ruas dikalikan dengan [latex]\large \frac{1}{tan^{2}\: \alpha }[/latex] )

[latex]\large \Rightarrow \frac{sec^{2}\: \alpha }{tan^{2}\: \alpha }-\frac{tan^{2}\: \alpha}{tan^{2}\: \alpha}=\frac{1}{tan^{2}\: \alpha}[/latex]

[latex]\large \Rightarrow \frac{1}{sin^{2}\alpha } – 1=\frac{1}{tan^{2}\alpha }[/latex]

[latex]\large \Rightarrow cosec^{2}\alpha -1=cotan^{2}\alpha[/latex]

[latex]\large \Rightarrow cosec^{2}\alpha =1+cotan^{2}\alpha[/latex]

4. [latex]\large tan \: \alpha =\frac{sin\: \alpha }{cos\: \alpha }[/latex]

5. [latex]\large cotan \: \alpha =\frac{cos\: \alpha }{tan\: \alpha }[/latex]

6. [latex]\large sec\: \alpha =\frac{1 }{tan\: \alpha }[/latex]

7. [latex]\large cosec\: \alpha =\frac{1 }{sin\: \alpha }[/latex]

8.[latex]\large tan\: \alpha =\frac{1 }{cotan\: \alpha }[/latex]

Contoh Soal
1. Buktikan identitas berikut
a. [latex]\large cos \alpha \: cotan \: \alpha = cosec \alpha – sin \alpha[/latex]

b. [latex]\large \frac{1+cot^{2\alpha }}{cot \: \alpha \: sec^{2}\alpha }= cot \: \alpha[/latex]

2. Sederhanakanlah bentuk berikut dengan menggunakan identitas trigonometri.

a. [latex]\large tan \: A – \frac{sec^{2}A}{tan \: A}[/latex]

b. [latex]\large \left ( cosec \: B – cotan \: B \right ) \left ( 1+ cos \: B \right )[/latex]

c. [latex]\large \frac{sin\: C}{1+ cos\: C}+\frac{sin\: C}{1- cos\: C}[/latex]
Jawab:
1.Pembuktian

a. [latex]\large cos \alpha \: cotan \: \alpha = cosec \alpha – sin \alpha[/latex]

[latex]\large \Rightarrow cos \: \alpha \cdot \frac{cos \: \alpha }{sin \: \alpha }[/latex]

[latex]\large \Rightarrow \frac{cos^{2} \: \alpha }{sin \: \alpha }[/latex]

[latex]\large \Rightarrow \frac{1-sin^{2} \: \alpha }{sin \: \alpha }[/latex]

[latex]\large \Rightarrow \frac{1}{sin \: \alpha }- \frac{sin^{2}\alpha }{sin \: \alpha }[/latex]

[latex]\large \Rightarrow cosec \: \alpha – sin \: \alpha[/latex]

Jadi, terbukti bahwa [latex]\large cos \alpha \: cotan \: \alpha = cosec \alpha – sin \alpha[/latex]

b. [latex]\large \frac{1+cot^{2\alpha }}{cot \: \alpha \: sec^{2}\alpha }= cot \: \alpha[/latex]

[latex]\large \Rightarrow \frac{cosec^{2}\alpha }{\frac{cos\: \alpha }{sin \: \alpha}\cdot \frac{1}{cos^{2}\alpha }}[/latex]

[latex]\large \Rightarrow \frac{\frac{1}{sin^{2}\alpha }}{\frac{cos\: \alpha }{sin \: \alpha}\cdot \frac{1}{cos^{2}\alpha }}[/latex]

[latex]\large \Rightarrow \frac{\frac{1}{sin^{2}\alpha }}{\frac{cos\: \alpha }{sin \: \alpha \: cos^{2}\alpha }}[/latex]

[latex]\large \Rightarrow \frac{1}{sin^{2}\alpha }\cdot \frac{sin\: \alpha \: cos^{2}\alpha }{cos\: \alpha }[/latex]

[latex]\large \Rightarrow \frac{cos\: \alpha }{sin \: \alpha }[/latex]

[latex]\large cot\: \alpha [/latex]

Jadi, terbukti bahwa [latex]\large \frac{1+cot^{2\alpha }}{cot \: \alpha \: sec^{2}\alpha }= cot \: \alpha[/latex]

2. Menyederhanakan:

a. [latex]\large tan \: A – \frac{sec^{2}A}{tan \: A}[/latex]

[latex]\large \Rightarrow \large tan \: A – \frac{1+tan^{2}A}{tan \: A}[/latex]

[latex]\large \Rightarrow \large \frac{tan^{2}A -(1+tan^{2}A)}{tan \: A}[/latex]

[latex]\large \Rightarrow \large \frac{tan^{2}A -1-tan^{2}A)}{tan \: A}[/latex]

[latex]\large \Rightarrow \large -\frac{1}{tan \: A}[/latex]

[latex]\large \Rightarrow \large cotan \: A[/latex]

b. [latex]\large \left ( cosec \: B – cotan \: B \right ) \left ( 1+ cos \: B \right )[/latex] [latex]\large \Rightarrow[/latex] Ubah ke dalam bentuk sin B dan cos B

[latex]\large \Rightarrow \frac{1}{sin\: B}+ \frac{1}{sin\: B}\cdot cos\: B- \frac{cos\: B}{sin\: B}-\frac{cos\: B}{sin\: B}\cdot cos\: B[/latex]

[latex]\large \Rightarrow \frac{1}{sin\: B}+ \frac{ cos\: B}{sin\: B}- \frac{cos\: B}{sin\: B}-\frac{cos^{2} B}{sin\: B}[/latex]

[latex]\large \Rightarrow \frac{1}{sin\: B}-\frac{cos^{2} B}{sin\: B}[/latex]

[latex]\large \Rightarrow \frac{1-cos^{2} B}{sin\: B}[/latex]

[latex]\large \Rightarrow \frac{sin^{2} B}{sin\: B}[/latex]

[latex]\large \Rightarrow \frac{1}{sin\: B}[/latex]

[latex]\large \Rightarrow cosec\: B[/latex]

c. [latex]\large \frac{sin\: C}{1+ cos\: C}+\frac{sin\: C}{1- cos\: C}[/latex]

[latex]\large \Rightarrow \frac{sin\: C (1-cos\: C)+sin \: (1+cos\: C)}{1-cos^{2} C}[/latex]

[latex]\large \Rightarrow \frac{sin\: C (1-cos\: C)+sin \: (1+cos\: C)}{sin^{2} C}[/latex]

[latex]\large \Rightarrow \frac{1-cos\: C+1+cos\: C}{sin\: C}[/latex]

[latex]\large \Rightarrow \frac{2}{sin\: C}[/latex]

[latex]\large \Rightarrow 2\frac{1}{sin\: C}[/latex]

[latex]\large \Rightarrow 2cosec\: C[/latex]