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 $a^{2}+b^{2}\geq c^{2}$

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

contoh

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

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

Jawaban

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