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
• $3ta{n}^2(x-45^{\circ })-4\sqrt{3}tan(x-45^{\circ })+3=0$
• 3 cos x + 4 sin x = 5

Jawaban

$$\begin{array}{cc}\ast sin(x-30^{\circ })& =\frac{1}{2}\\ sin(x-30^{\circ })& =sin30^{\circ }\\ \ast \ast x-30^{\circ }& =30^{\circ }+k360^{\circ }\\ x& =(30^{\circ }+30^{\circ })+k360^{\circ }\\ x& =60^{\circ }+k360^{\circ }\\ k& =0x=60^{\circ }\\ \ast \ast x-30^{\circ }& =(180^{\circ }-30^{\circ })+k360^{\circ }\\ x-30^{\circ }& =150^{\circ }+k360^{\circ }\\ x& =(150^{\circ }+30^{\circ })+k360^{\circ }\\ x& =180^{\circ }+k360^{\circ }\\ k& =0x=180^{\circ }\\ \ast cosx& =2sin4xcos4x\\ cosx& =sin8x\\ cosx& =cos(90^{\circ }-8x)\\ \ast \ast x& =90^{\circ }-8x+k360^{\circ }\\ 9x& =90^{\circ }+k360^{\circ }\\ x& =10^{\circ }+k40^{\circ }\\ k& =0x=10^{\circ }\\ k& =1x=50^{\circ }\\ k& =2x=90^{\circ }\\ k& =3x=130^{\circ }\\ k& =4x=170^{\circ }\\ k& =5x=210^{\circ }\\ k& =6x=250^{\circ }\\ k& =7x=290^{\circ }\\ k& =8x=330^{\circ }\\ \ast \ast x& =-(90^{\circ }-8x)+k360^{\circ }\\ x& =-90^{\circ }+8x+k360^{\circ }\\ -7x& =-90^{\circ }+k360^{\circ }\\ x& =(\frac{90}{7}{)}^{\circ }-k(\frac{360}{7}{)}^{\circ }\\ k& =0x=(\frac{90}{7}{)}^{\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 }\\ \ast \ast \ast x-45^{\circ }& =30^{\circ }+k180^{\circ }\\ x& =(30^{\circ }+45^{\circ })+k180^{\circ }\\ 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 }\\ \ast \ast \ast x-45^{\circ }& =60^{\circ }+k180^{\circ }\\ x& =(60^{\circ }+45^{\circ })+k180^{\circ }\\ x& =105^{\circ }+k180^{\circ }\\ k& =0x=105^{\circ }\\ k& =1x=285^{\circ }\\ \ast 3cosx+4sinx& =5\\ k& =\sqrt{3^2+4^2}=5\\ tan\alpha & =\frac{4}{3}\\ \alpha & =37^{\circ }\\ kcos(x-\alpha )& =c\\ 5cos(x-37^{\circ })& =5\\ cos(x-37^{\circ })& =1\\ cos(x-37^{\circ })& =cos{0}^{\circ }\\ \ast \ast x-37^{\circ }& =0^{\circ }+k360^{\circ }\\ x& =(0^{\circ }+37^{\circ })+k360^{\circ }\\ x& =37^{\circ }+k360^{\circ }\\ k& =0x=37^{\circ }\end{array}$$