Soal-Soal Matematika/Persamaan dan pertidaksamaan trigonometri

Sistem persamaan

Jika $sinx=sin\alpha$, maka $x=\alpha +k\cdot 360^{\circ }\text{ atau }x=(180^{\circ }-\alpha )+k\cdot 360^{\circ }$ serta $x=\alpha +k\cdot 2\pi \text{ atau }x=(2\pi -\alpha )+k\cdot 2\pi$

Jika $cosx=cos\alpha$, maka $x=\pm \alpha +k\cdot 360^{\circ }$ serta $x=\pm \alpha +k\cdot 2\pi$

Jika $tanx=tan\alpha$, maka $x=\alpha +k\cdot 180^{\circ }$ serta $x=\alpha +k\cdot \pi$

Persamaan $a\cos x+b\sin x=c$ dapat diubah menjadi $k\cos (x-\alpha )=c$, maka $k=\sqrt{a^2+b^2}$, $\tan \alpha =\frac{b}{a}$ serta $a^2+b^2\ge c^2$

Perhatian: $sin(x-30^{\circ })\ne sin(x-30{)}^{\circ }$

contoh

1. Dengan batas-batas 0° ≤ x ≤ 360°. tentukan:

• $sin(x-30^{\circ })=1$
• cos x = 2 sin 4x cos 4x
• $3ta{n}^2(x-45^{\circ })-4\sqrt{3}tan(x-45^{\circ })+3=0$

Jawaban

$$\begin{array}{cc}\ast sin(x-30^{\circ })& =1\\ sin(x-30^{\circ })& =sin90^{\circ }\\ x-30^{\circ }& =90^{\circ }\\ x& =90^{\circ }+30^{\circ }\\ x& =120^{\circ }\\ \ast \ast x& =120^{\circ }+k360^{\circ }\\ k& =0x=120^{\circ }\\ \ast \ast x& =(180-120{)}^{\circ }+k360^{\circ }\\ x& =60^{\circ }+k6360^{\circ }\\ k& =0x=60^{\circ }\\ \ast cosx& =2sin4xcos4x\\ cosx& =sin8x\\ cosx& =cos(90^{\circ }-8x)\\ x& =90^{\circ }-8x\\ 9x& =90^{\circ }\\ x& =10^{\circ }\\ \ast \ast x& =10^{\circ }+k360^{\circ }\\ k& =0x=10^{\circ }\\ \ast \ast x& =-10^{\circ }+k360^{\circ }\\ k& =0x=350^{\circ }\\ \ast 3ta{n}^2(x-45^{\circ })-4\sqrt{3}tan(x-45^{\circ })+3& =0\\ (3tan(x-45^{\circ })-\sqrt{3})(tan(x-45^{\circ })-\sqrt{3})& =0\\ tan(x-45^{\circ })& =\frac{\sqrt{3}}{3}\text{atau}tan(x-45^{\circ })=\sqrt{3}\\ \ast \ast tan(x-45^{\circ })& =\frac{\sqrt{3}}{3}\\ tan(x-45^{\circ })& =tan30^{\circ }\\ x-45^{\circ }& =30^{\circ }\\ x& =30^{\circ }+45^{\circ }\\ x& =75^{\circ }\\ \ast \ast \ast x& =75^{\circ }+k180^{\circ }\\ k& =0x=75^{\circ }\\ k& =1x=255^{\circ }\\ \ast \ast tan(x-45^{\circ })& =\sqrt{3}\\ tan(x-45^{\circ })& =tan60^{\circ }\\ x-45^{\circ }& =60^{\circ }\\ x& =60^{\circ }+45^{\circ }\\ x& =105^{\circ }\\ \ast \ast \ast x& =105^{\circ }+k180^{\circ }\\ k& =0x=105^{\circ }\\ k& =1x=285^{\circ }\\ \end{array}$$