Soal-Soal Matematika/Persamaan dan pertidaksamaan trigonometri

Sistem persamaan

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

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

Jawaban

$${\begin{aligned}*sin(x-30^{\circ })&={\frac {1}{2}}\\sin(x-30^{\circ })&=sin30^{\circ }\\**x-30^{\circ }&=30^{\circ }+k360^{\circ }\\x&=(30^{\circ }+30^{\circ })+k360^{\circ }\\x&=60^{\circ }+k360^{\circ }\\k&=0x=60^{\circ }\\**x-30^{\circ }&=(180^{\circ }-30^{\circ })+k360^{\circ }\\x-30^{\circ }&=150^{\circ }+k360^{\circ }\\x&=(150^{\circ }+30^{\circ })+k360^{\circ }\\x&=180^{\circ }+k360^{\circ }\\k&=0x=180^{\circ }\\*cosx&=2sin4xcos4x\\cosx&=sin8x\\cosx&=cos(90^{\circ }-8x)\\**x&=90^{\circ }-8x+k360^{\circ }\\9x&=90^{\circ }+k360^{\circ }\\x&=10^{\circ }+k40^{\circ }\\k&=0x=10^{\circ }\\k&=1x=50^{\circ }\\k&=2x=90^{\circ }\\k&=3x=130^{\circ }\\k&=4x=170^{\circ }\\k&=5x=210^{\circ }\\k&=6x=250^{\circ }\\k&=7x=290^{\circ }\\k&=8x=330^{\circ }\\**x&=-(90^{\circ }-8x)+k360^{\circ }\\x&=-90^{\circ }+8x+k360^{\circ }\\-7x&=-90^{\circ }+k360^{\circ }\\x&=({\frac {90}{7}})^{\circ }-k({\frac {360}{7}})^{\circ }\\k&=0x=({\frac {90}{7}})^{\circ }\\*3tan^{2}(x-45^{\circ })-4{\sqrt {3}}tan(x-45^{\circ })+3&=0\\(3tan(x-45^{\circ })-{\sqrt {3}})(tan(x-45^{\circ })-{\sqrt {3}})&=0\\tan(x-45^{\circ })&={\frac {\sqrt {3}}{3}}{\text{ atau }}tan(x-45^{\circ })={\sqrt {3}}\\**tan(x-45^{\circ })&={\frac {\sqrt {3}}{3}}\\tan(x-45^{\circ })&=tan30^{\circ }\\***x-45^{\circ }&=30^{\circ }+k180^{\circ }\\x&=(30^{\circ }+45^{\circ })+k180^{\circ }\\x&=75^{\circ }+k180^{\circ }\\k&=0x=75^{\circ }\\k&=1x=255^{\circ }\\**tan(x-45^{\circ })&={\sqrt {3}}\\tan(x-45^{\circ })&=tan60^{\circ }\\***x-45^{\circ }&=60^{\circ }+k180^{\circ }\\x&=(60^{\circ }+45^{\circ })+k180^{\circ }\\x&=105^{\circ }+k180^{\circ }\\k&=0x=105^{\circ }\\k&=1x=285^{\circ }\\*3cosx+4sinx&=5\\k&={\sqrt {3^{2}+4^{2}}}=5\\tan\alpha &={\frac {4}{3}}\\\alpha &=37^{\circ }\\kcos(x-\alpha )&=c\\5cos(x-37^{\circ })&=5\\cos(x-37^{\circ })&=1\\cos(x-37^{\circ })&=cos0^{\circ }\\**x-37^{\circ }&=0^{\circ }+k360^{\circ }\\x&=(0^{\circ }+37^{\circ })+k360^{\circ }\\x&=37^{\circ }+k360^{\circ }\\k&=0x=37^{\circ }\\\end{aligned}}$$