Trigonometrinės lygtys
Užduotis. Sprendimas

Trigonometrinės lygtys Uždavinys

Užduotis. Duotas reiškinys $4\sin (62.5^\circ+{\cfrac{x}{2}})\cos (27.5^\circ- {\cfrac{x}{2}})$
1. Apskaičiuokite šio reiškinio reikšmę su kintamojo $x$ reikšme, lygia $55^\circ$
2. Pagrįskite, kad duotasis reiškinys tapačiai lygus reiškiniui $4\sin^2 ({\cfrac{x}{2}}+62.5^\circ)$
3. Išspręskite lygtį $4\sin (62.5^\circ+{\cfrac{x}{2}})\cos (27.5^\circ- {\cfrac{x}{2}})=3$
4. Rasti lygties $4\sin (62.5^\circ+{\cfrac{x}{2}})\cos (27.5^\circ- {\cfrac{x}{2}})=3$ sprendinį (laipsniais), priklausantį intervalui $(0;180^\circ)$

Sprendimas.
1.
$x=\color{#ff7f0e}{55^\circ}$
$4\sin (62.5^\circ+{\cfrac{\color{#ff7f0e}{55^\circ}}{2}})\cos (27.5^\circ- {\cfrac{\color{#ff7f0e}{55^\circ}}{2}})$
$4\sin (62.5^\circ+27.5^\circ)\cos (27.5^\circ- 27.5^\circ)$
$4\sin(90^\circ)\cos(0^\circ)=4 \cdot 1 \cdot 1 = 4$
2.
Pritaikome formulę $\color{#1f77b4}{\cos(90^\circ-\alpha)=\sin(\alpha)}$
$27.5^\circ=90^\circ-62.5^\circ$
$\cos (27.5^\circ- {\cfrac{x}{2}})=\cos(90-(62.5^\circ+\cfrac{x}{2}))$
$\cos (27.5^\circ- {\cfrac{x}{2}})=\sin({\cfrac{x}{2}}+62.5^\circ)$
$4\sin (62.5^\circ+{\cfrac{x}{2}})\cos (27.5^\circ- {\cfrac{x}{2}})=$$4\sin^2 ({\cfrac{x}{2}}+62.5^\circ)$
3.
$4\sin (62.5^\circ+{\cfrac{x}{2}})\cos (27.5^\circ- {\cfrac{x}{2}})=3$
$4\sin^2 ({\cfrac{x}{2}}+62.5^\circ)=3$
Pritaikome formulę $\color{#1f77b4}{2 \sin^2 (\alpha)=1-\cos(2\alpha)}$
$2 ({1-\cos(x+125^\circ)})=3$
$2-2\cos(x+125^\circ)=3$
$\cos(x+125^\circ)=-\cfrac{1}{2}$
Pritaikome formules
$\color{#1f77b4}{\cos(x)=\alpha}$
$\color{#1f77b4}{x=\pm\arccos (\alpha)+360^\circ k, k \in \mathbb Z}$
$125^\circ+x=\pm \arccos (-\cfrac{1}{2})+360^\circ k, k \in \mathbb Z $
$125^\circ+x=\pm 120^\circ + 360^\circ k, k \in \mathbb Z $
Atsakymas $x=\pm 120^\circ – 125^\circ+ 360^\circ k, k \in \mathbb Z $
4.
Kai $k=0$, tai $x=\pm120^\circ-125^\circ$
$x_1=-120^\circ-125^\circ=-245^\circ$
$x_2=120^\circ-125^\circ=-5^\circ$
Kai $k=1$, tai $x=\pm120^\circ-125^\circ+360^\circ$
$x_3=-120^\circ+235^\circ=115^\circ$
$x_4=120^\circ+235^\circ=355^\circ$
Atsakymas $115^\circ$