Тригонометрийн адилтгалууд

Үндсэн адилтгалууд $$\sin^2\alpha+\cos^2\alpha=1$$ $$1+\mbox{ctg}^2\alpha=\dfrac1{\sin^2\alpha}$$ $$1+\mbox{tg}^2\alpha=\dfrac1{\cos^2\alpha}$$

1. $\alpha$ өнцгийн хувьд дараах утгууд зөв үү?
   1. $\sin\alpha=-1,\ \cos\alpha=\dfrac12$;
   2. $\sin\alpha=\dfrac35,\ \cos\alpha=-\dfrac45$;
   3. $\sin\alpha=\dfrac{\sqrt2}2,\ \cos\alpha=\dfrac{\sqrt2}2$;
2. $\sin\alpha$-г ол.
   1. $\cos\alpha=\dfrac14,\ 0< \alpha< \dfrac\pi2$;
   2. $\cos\alpha=-\dfrac13,\ \pi< \alpha< \dfrac{3\pi}2$;
3. $\cos\alpha$-г ол.
   1. $\sin\alpha=0.8,\ \dfrac\pi2< \alpha< \pi$;
   2. $\sin\alpha=-0.6,\ \dfrac{3\pi}2< \alpha< 2\pi$;
4. Илэрхийллийг хялбарчил.
   1. $1-\sin^2\alpha$;
   2. $1-\cos^2\alpha$;
   3. $\sin^2\alpha-1$;
   4. $\cos^2\alpha-1$;
   5. $(1-\sin\alpha)(1+\sin\alpha)$;
   6. $(\cos\alpha-1)(\cos\alpha+1)$;
   7. $\cos^2\alpha-\sin^2\alpha+1$;
   8. $1+\sin^2\alpha-\cos^2\alpha$;
   9. $1-\sin^2\alpha-\cos^2\alpha$;
5. Илэрхийллийг хялбарчил.
   1. $\dfrac{1-\sin^2\alpha}{\cos^2\alpha}$;
   2. $\dfrac{\cos^2\alpha-1}{\sin^2\alpha}$;
   3. $\dfrac{\sin^2\alpha}{1+\cos\alpha}$;
   4. $\dfrac{\cos^2\alpha}{\sin\alpha-1}$;

Нийлбэрийн синус ба косинус $$\sin(\alpha\pm\beta)=\sin\alpha\cos\beta\pm\cos\alpha\sin\beta$$ $$\cos(\alpha\pm\beta)=\cos\alpha\cos\beta\mp\sin\alpha\sin\beta$$

1. Дээрх томьёонуудыг ашиглан дараах томьёонуудыг батал.
   1. $\sin(\pi-\alpha)=\sin\alpha$
   2. $\cos(\pi-\alpha)=-\cos\alpha$
   3. $\sin(\pi+\alpha)=-\sin\alpha$
   4. $\cos(\pi+\alpha)=-\cos\alpha$
   5. $\cos\left(\dfrac\pi2+\alpha\right)=-\sin\alpha$
   6. $\sin\left(\dfrac\pi2+\alpha\right)=\cos\alpha$
   7. $\sin\left(\dfrac\pi2-\alpha\right)=\cos\alpha$
   8. $\cos\left(\dfrac\pi2-\alpha\right)=\sin\alpha$
2. Дараах адилтгалуудыг батал.
   1. $\sin\left(\dfrac\pi2+\alpha\right)=\cos\alpha$;
   2. $\sin\left(\dfrac{3\pi}2-\alpha\right)=-\cos\alpha$;
   3. $\sin\left(\pi-\alpha\right)=\sin\alpha$;
   4. $\sin\left(\dfrac{3\pi}2+\alpha\right)=-\cos\alpha$;
3. Тооцоол.
   1. $\sin20^\circ\cos10^\circ+\cos20^\circ\sin10^\circ$;
   2. $\sin\dfrac\pi5\cos\dfrac{4\pi}5+\cos\dfrac\pi5\sin\dfrac{4\pi}5$;
   3. $\cos80^\circ\sin10^\circ+\sin80^\circ\cos10^\circ$; 
   4. $\cos\dfrac{3\pi}8\sin\dfrac\pi8+\cos\dfrac\pi8\sin\dfrac{3\pi}8$;
4. Дараах утгуудыг ол.
   1. $\sin75^\circ$
   2. $\sin105^\circ$
   3. $\cos165^\circ$
   4. $\cos195^\circ$
5. Илэрхийллийг хялбарчил.
   1. $\dfrac{\sqrt3}2\sin\alpha-\dfrac12\cos\alpha$
   2. $\dfrac{\sqrt2}2(\cos\alpha-\sin\alpha)$
   3. $\dfrac{\sqrt2}2(\sin\alpha+\cos\alpha)$
   4. $\dfrac12\sin\alpha+\dfrac{\sqrt3}2\cos\alpha$
6. Тооцоол.
   1. Хэрэв $\sin\alpha=\dfrac12$, $\cos\beta=\dfrac13$ ба $0< \alpha< \dfrac\pi2$, $0< \beta< \dfrac\pi2$ бол $\sin(\alpha+\beta)$-г
   2. Хэрэв $\cos\alpha=-0.2$, $\cos\beta=-0.1$ ба $\dfrac\pi2< \alpha< \pi$, $\pi< \beta< \dfrac{3\pi}2$ бол $\sin(\alpha-\beta)$-г тооцоол.

Синус, косинусын нийлбэрийн томьёо  $$\sin\alpha+\sin\beta=2\sin\dfrac{\alpha+\beta}2\cos\dfrac{\alpha-\beta}2$$ $$\sin\alpha-\sin\beta=2\sin\dfrac{\alpha-\beta}2\cos\dfrac{\alpha+\beta}2$$ $$\cos\alpha+\cos\beta=2\cos\dfrac{\alpha+\beta}2\cos\dfrac{\alpha-\beta}2$$ $$\cos\alpha-\cos\beta=-2\sin\dfrac{\alpha+\beta}2\sin\dfrac{\alpha-\beta}2$$

1. Нийлбэр хэлбэрт байгаа дараах илэрхийллүүдийг үржвэр хэлбэрт бич.
   1. $\sin20^\circ+\sin10^\circ$
   2. $\sin60^\circ-\sin30^\circ$
   3. $\cos70^\circ+\cos20^\circ$
   4. $\cos80^\circ-\cos30^\circ$
   5. $\cos\dfrac\pi5-\cos\dfrac\pi4$
   6. $\sin\dfrac\pi{14}+\sin\dfrac\pi3$
   7. $\sin\dfrac\pi3-\sin\dfrac\pi4$
   8. $\cos\dfrac\pi{10}+\cos\dfrac\pi5$
2. Нийлбэр хэлбэрт байгаа дараах илэрхийллүүдийг үржвэр хэлбэрт бич.
   1. $\sin\alpha+\sin3\alpha$
   2. $\cos3\alpha-\cos\alpha$
   3. $\cos7\alpha+\cos\alpha$
   4. $\sin3\alpha-\sin5\alpha$
   5. $\sin\alpha+\cos\alpha$
   6. $\cos\alpha-\sin\alpha$
3. Дараах тэнцэтгэлүүдийг батал.
   1. $\sin50^\circ+\sin10^\circ-\cos20^\circ=0$
   2. $\cos48^\circ+\sin18^\circ-\cos12^\circ=0$
4. Тооцоол.
   1. $\cos\dfrac{5\pi}{12}+\cos\dfrac{\pi}{12}$
   2. $\cos\dfrac{7\pi}{12}-\cos\dfrac{\pi}{12}$
   3. $\sin\dfrac{5\pi}{12}+\sin\dfrac{\pi}{12}$
   4. $\sin\dfrac{7\pi}{12}-\sin\dfrac{\pi}{12}$
5. Дараах тэнцэтгэлүүдийг батал.
   1. $\cos\dfrac{5\pi}{12}+\cos\dfrac{7\pi}{12}=0$
   2. $\cos\dfrac{9\pi}{14}+\cos\dfrac{5\pi}{14}=0$
   3. $\sin\dfrac{3\pi}{5}-\sin\dfrac{2\pi}{5}=0$
   4. $\sin\dfrac{3\pi}{10}-\sin\dfrac{7\pi}{10}=0$
6. Тооцоол.
   1. $\cos75^\circ\cdot\cos105^\circ$
   2. $\cos\dfrac{75^\circ}2\cdot\cos\dfrac{15^\circ}2$
   3. $\sin75^\circ\cdot\sin15^\circ$
   4. $\sin105^\circ\cdot\cos15^\circ$

Давхар өнцгийн томьёо $$\sin2\alpha=2\sin\alpha\cos\alpha$$ $$\cos2\alpha=\cos^2\alpha-\sin^2\alpha$$ $$\cos2\alpha=2\cos^2\alpha-1$$ $$\cos2\alpha=1-2\sin^2\alpha$$

1. Тооцоол.
   1. $2\sin15^\circ\cos15^\circ$ 
   2. $4\sin22^\circ30'\cos22^\circ30'$
   3. $5\sin\dfrac\pi{12}\cos\dfrac\pi{12}$
   4. $4\cos(-15^\circ)\sin(-15^\circ)$
2. $\sin2\alpha$-г ол.
   1. $\sin\alpha=\dfrac12,\ 0< \alpha< \dfrac\pi2$
   2. $\cos\alpha=-\dfrac13,\ \dfrac\pi2< \alpha< \pi$
3. Дараах илэрхийллүүдийн утгыг ол.
   1. $\cos^215^\circ-\sin^215^\circ$
   2. $\sin^215^\circ-\cos^215^\circ$
   3. $\cos^220^\circ-\sin^220^\circ$
   4. $(\sin\alpha+\cos\alpha)(\sin\alpha-\cos\alpha)$
   5. $\sin^2\dfrac\pi8-\cos^2\dfrac\pi8$
   6. $2\sin50^\circ\sin40^\circ$
   7. $\cos^215^\circ-\cos^275^\circ$
4. Илэрхийллүүдийг хялбарчил.
   1. $\sin\alpha\cos\alpha\cos2\alpha$ 
   2. $\cos^4\alpha-\sin^4\alpha$
   3. $\dfrac{(\sin\alpha+\cos\alpha)^2}{1+\sin2\alpha}$, $\alpha\neq-\dfrac\pi4+\pi k,\ k\in \mathbf{Z}$
   4. $\dfrac{\cos2\alpha}{\sin\alpha-\cos\alpha}$, $\alpha\neq\dfrac\pi4+\pi k,\ k\in \mathbf{Z}$
   5. $2\cos^2\alpha-\cos2\alpha$
   6. $\cos2\alpha+2\sin^2\alpha$ 
5. $\sin\dfrac\alpha2$-г ол.
   1. $\cos\alpha=\dfrac13,\ 0< \alpha< \dfrac\pi2$
   2. $\sin\alpha=\dfrac35,\ \dfrac\pi2< \alpha< \pi$
6. $\cos\dfrac\alpha2$-г ол.
   1. $\sin\alpha=-\dfrac13,\ \pi< \alpha< \dfrac{3\pi}2$
   2. $\cos\alpha=-\dfrac{12}{13},\ \dfrac\pi2< \alpha< \pi$
7. Илэрхийллийг хялбарчил.
   1. $2\sin^2\dfrac\alpha2+\cos\alpha$
   2. $2\cos^2\dfrac\alpha2-\cos\alpha$
   3. $4\sin^2\dfrac\alpha2+2\cos\alpha+3$
   4. $4\cos^2\dfrac\alpha2-2\cos\alpha+3$

Синус, косинусын үржвэр $$\sin\alpha\cos\beta=\dfrac12(\sin(\alpha+\beta)+\sin(\alpha-\beta))$$ $$\cos\alpha\cos\beta=\dfrac12(\cos(\alpha+\beta)+\cos(\alpha-\beta))$$ $$\sin\alpha\sin\beta=\dfrac12(\cos(\alpha-\beta)-\cos(\alpha+\beta))$$

1. Илэрхийллийг нийлбэр хэлбэрт бич.
   1. $\cos3\alpha\cos\alpha$
   2. $\cos\alpha\cos2\alpha$
   3. $\sin5\alpha\sin3\alpha$
   4. $\sin2\alpha\sin3\alpha$
   5. $\sin4\alpha\cos2\alpha$
   6. $\sin\alpha\cos4\alpha$
2. Илэрхийллийн утгыг ол.
   1. $\sin\dfrac{11\pi}{24}\sin\dfrac{5\pi}{24}$
   2. $\cos\dfrac{13\pi}{24}\cos\dfrac{7\pi}{24}$
   3. $\sin\dfrac{7\pi}{24}\cos\dfrac{\pi}{24}$
   4. $\cos63^\circ\cos27^\circ-\sin12^\circ\sin48^\circ$
   5. $\cos\dfrac{11\pi}{56}\cos\dfrac{3\pi}{56}-\sin\dfrac{11\pi}{42}\sin\dfrac{17\pi}{42}$

Нийлбэрийн тангенс $$\mbox{tg}(\alpha+\beta)=\dfrac{\mbox{tg}\alpha+\mbox{tg}\beta}{1-\mbox{tg}\alpha\mbox{tg}\beta}$$ $$\mbox{tg}(\alpha-\beta)=\dfrac{\mbox{tg}\alpha-\mbox{tg}\beta}{1+\mbox{tg}\alpha\mbox{tg}\beta}$$ $$\mbox{tg}2\alpha=\dfrac{2\mbox{tg}\alpha}{1-\mbox{tg}^2\alpha}$$ $$\mbox{tg}\dfrac\alpha2=\dfrac{\sin\alpha}{1+\cos\alpha}$$ $$\mbox{tg}\dfrac\alpha2=\dfrac{1-\cos\alpha}{\sin\alpha}$$

1. Илэрхийллийн утгыг ол.
   1. $\mbox{tg}(60^\circ+45^\circ)$
   2. $\mbox{tg}\left(\dfrac\pi6+\dfrac\pi4\right)$
   3. $\mbox{tg}(60^\circ-45^\circ)$
   4. $\mbox{tg}\left(\dfrac\pi4-\dfrac\pi6\right)$
2. Хэрэв $\mbox{tg}\alpha=\dfrac53$, $\mbox{tg}\beta=\dfrac25$ бол $\mbox{tg}(\alpha+\beta)$ ба $\mbox{tg}(\alpha-\beta)$ илэрхийллүүдийн утгыг ол.
3. Илэрхийллийн утгыг ол.
   1. $\dfrac{\mbox{tg}39^\circ+\mbox{tg}6^\circ}{1-\mbox{tg}39^\circ\mbox{tg}6^\circ}$
   2. $\dfrac{\mbox{tg}72^\circ-\mbox{tg}12^\circ}{1+\mbox{tg}72^\circ\mbox{tg}12^\circ}$
   3. $\dfrac{37^\circ+\mbox{tg}23^\circ}{1-\mbox{tg}37^\circ\mbox{tg}23^\circ}$
   4. $\dfrac{\mbox{tg}54^\circ-\mbox{tg}24^\circ}{1+\mbox{tg}54^\circ\mbox{tg}24^\circ}$
4. $\mbox{tg}2\alpha$-г ол.
   1. Хэрэв $\mbox{tg}\alpha=\dfrac17$ бол
   2. Хэрэв $\mbox{tg}\alpha=-\dfrac14$ бол
   3. Хэрэв $\mbox{tg}\alpha=3$ бол
   4. Хэрэв $\mbox{tg}\alpha=-4$ бол
   5. Хэрэв $\sin\alpha=\dfrac35$, $\cos\alpha=\dfrac45$ бол
   6. Хэрэв $\sin\alpha=-\dfrac5{13}$, $\cos\alpha=\dfrac{12}{13}$ бол