Trigonometri #3

3.       Hubungan antara sudut lancip dan komplemennya

$latex sin\alpha =\frac{a}{c}=cos(90^\circ -\alpha )$

$latex cos\alpha =\frac{b}{c}=sin(90^\circ -\alpha )$

$latex tan\alpha =\frac{a}{b}=cot(90^\circ -\alpha )$

$latex cot\alpha =\frac{b}{a}=tan(90^\circ -\alpha )$

$latex sec\alpha =\frac{c}{b}=csc(90^\circ -\alpha )$

$latex csc\alpha =\frac{c}{a}=sec(90^\circ -\alpha )$

Contoh Soal

1.        Diketahui $latex tan(\alpha +x)=cot(\alpha -x)$. Tentukan sudut $latex \alpha$.

Jawaban:

$latex \begin{array}{rcl}tan(\alpha +x)&=&cot(\alpha -x)\\tan(\alpha +x)&=&tan\Big(90^\circ -(\alpha -x)\Big)\\\alpha +x&=&90^\circ -(\alpha -x)\\2\alpha &=&90^\circ \\\alpha &=&45^\circ \end{array}$

2.      Untuk sudut lancip $latex x$ diketahui $latex tan\ x$ adalah dua kali kosinus komplemennya. Carilah nilai $latex x$.

Jawaban:

$latex \begin{array}{rcl} tanx&=&2\ cos(90^\circ -x)\\tanx&=&2\ sinx\\\frac{sinx}{cosx}&=&2\ sinx\\sinx &=&2\ sinx\ cosx\\2\ sinx\ cosx-sinx&=&0\\sinx(2cosx-1)&=&0\\sinx=0&atau&2cosx-1=0\\x=0^\circ &atau&x=60^\circ \end{array}$

3.      Buktikan $latex \frac{2tan\varphi }{1+tan^2\varphi }=2\ sin\varphi \ sin(90^\circ -\varphi )$.

. Jawaban:

$latex \begin{array}{rcl} \frac{2\ tan\varphi }{1+tan^2\varphi }&=&\frac{2\ tan\varphi }{sec^2\varphi }\\\\&=&2\ \frac{sin\varphi }{cos\varphi }\ cos^2\varphi \\\\&=&2\ sin\varphi \ cos\varphi \\\\&=&2\ sin\varphi \ sin(90^\circ -\varphi )\end{array}$

Terbukti

Adjie Gumarang Pujakelana, 2013

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