Levi Rizki Saputra Notes

Sudut 75 Derajat

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# Penjumlahan 45 + 30

sin⁡(75°)=sin⁡(45°+30°)=sin⁡(45°)cos⁡(30°)+cos⁡(45°)sin⁡(30°)=22⋅32+22⋅12=24(3+1)cos⁡(75°)=cos⁡(45°+30°)=cos⁡(45°)cos⁡(30°)−sin⁡(45°)sin⁡(30°)=22⋅32−22⋅12=24(3−1)tan⁡(75°)=tan⁡(45°+30°)=tan⁡(45°)+tan⁡(30°)1−tan⁡(45°)⋅tan⁡(30°)=1+331−1⋅33=(3+33)(3−33)=3+33−3×3+33+3=32+2⋅3⋅3+(3)232−(3)2=9+63+39−3=12+636=6(2+3)6=2+3\begin{aligned} \sin( 75\degree ) & =\sin( 45\degree +30\degree )\\ & =\sin( 45\degree )\cos( 30\degree ) +\cos( 45\degree )\sin( 30\degree )\\ & =\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} +\frac{\sqrt{2}}{2} \cdot \frac{1}{2}\\ & =\frac{\sqrt{2}}{4}\left(\sqrt{3} +1\right)\\ \cos( 75\degree ) & =\cos( 45\degree +30\degree )\\ & =\cos( 45\degree )\cos( 30\degree ) -\sin( 45\degree )\sin( 30\degree )\\ & =\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} -\frac{\sqrt{2}}{2} \cdot \frac{1}{2}\\ & =\frac{\sqrt{2}}{4}\left(\sqrt{3} -1\right)\\ \tan( 75\degree ) & =\tan( 45\degree +30\degree )\\ & =\frac{\tan( 45\degree ) +\tan( 30\degree )}{1-\tan( 45\degree ) \cdot \tan( 30\degree )}\\ & =\frac{1+\frac{\sqrt{3}}{3}}{1-1\cdot \frac{\sqrt{3}}{3}}\\ & =\frac{\left(\frac{3+\sqrt{3}}{3}\right)}{\left(\frac{3-\sqrt{3}}{3}\right)}\\ & =\frac{3+\sqrt{3}}{3-\sqrt{3}} \times \frac{3+\sqrt{3}}{3+\sqrt{3}}\\ & =\frac{3^{2} +2\cdot 3\cdot \sqrt{3} +\left(\sqrt{3}\right)^{2}}{3^{2} -\left(\sqrt{3}\right)^{2}}\\ & =\frac{9+6\sqrt{3} +3}{9-3}\\ & =\frac{12+6\sqrt{3}}{6}\\ & =\frac{6\left( 2+\sqrt{3}\right)}{6}\\ & =2+\sqrt{3} \end{aligned}

# Pengurangan 90 - 15

sin⁡(75°)=sin⁡(90°−15°)=cos⁡(15°)=24(3+1)cos⁡(75°)=cos⁡(90°−15°)=sin⁡(15°)=24(3−1)tan⁡(75°)=tan⁡(90°−15°)=1tan⁡(15°)=12−3×2+32+3=2+322−(3)2=2+34−3=2+3\begin{aligned} \sin( 75\degree ) & =\sin( 90\degree -15\degree )\\ & =\cos( 15\degree )\\ & =\frac{\sqrt{2}}{4}\left(\sqrt{3} +1\right)\\ \cos( 75\degree ) & =\cos( 90\degree -15\degree )\\ & =\sin( 15\degree )\\ & =\frac{\sqrt{2}}{4}\left(\sqrt{3} -1\right)\\ \tan( 75\degree ) & =\tan( 90\degree -15\degree )\\ & =\frac{1}{\tan( 15\degree )}\\ & =\frac{1}{2-\sqrt{3}} \times \frac{2+\sqrt{3}}{2+\sqrt{3}}\\ & =\frac{2+\sqrt{3}}{2^{2} -\left(\sqrt{3}\right)^{2}}\\ & =\frac{2+\sqrt{3}}{4-3}\\ & =2+\sqrt{3} \end{aligned}

# Nilai tan Menurut sin dan cos

tan⁡(75°)=sin⁡(75°)cos⁡(75°)=24(3+1)24(3−1)=3+13−1×3+13+1=(3)2+2⋅1⋅3+12(3)2−12=3+23+13−1=4+232=2(2+3)2=2+3\begin{aligned} \tan( 75\degree ) & =\frac{\sin( 75\degree )}{\cos( 75\degree )}\\ & =\frac{\dfrac{\sqrt{2}}{4}\left(\sqrt{3} +1\right)}{\dfrac{\sqrt{2}}{4}\left(\sqrt{3} -1\right)}\\ & =\frac{\sqrt{3} +1}{\sqrt{3} -1} \times \frac{\sqrt{3} +1}{\sqrt{3} +1}\\ & =\frac{\left(\sqrt{3}\right)^{2} +2\cdot 1\cdot \sqrt{3} +1^{2}}{\left(\sqrt{3}\right)^{2} -1^{2}}\\ & =\frac{3+2\sqrt{3} +1}{3-1}\\ & =\frac{4+2\sqrt{3}}{2}\\ & =\frac{2\left( 2+\sqrt{3}\right)}{2}\\ & =2+\sqrt{3} \end{aligned}

# Kesimpulan

Nilai fungsi trigonometri 75°75\degree

sin⁡(75°)=24(3+1)cos⁡(75°)=24(3−1)tan⁡(75°)=2+3\begin{aligned} \sin( 75\degree ) & =\frac{\sqrt{2}}{4}\left(\sqrt{3} +1\right)\\ \cos( 75\degree ) & =\frac{\sqrt{2}}{4}\left(\sqrt{3} -1\right)\\ \tan( 75\degree ) & =2+\sqrt{3} \end{aligned}