antwoorden
Onderstaand overzicht volgt de nummering en de opgaven van de derde editie.
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1
Dit is een rechtstreekse toepassing van de definities en verwante hoeken, zie figuur 4.5 en paragraaf 4.2.1 en 4.2.2.
2
| a. $-\frac{\sqrt{2}}{2}$ |
c. $\frac{\sqrt{3}}{2}$ |
e. $2$ |
g. $-2$ |
| b. $-\frac{\sqrt{3}}{2}$ |
d. $\frac{1+\sqrt{3}}{2\sqrt{2}}$ |
f. $\frac{1}{\sqrt{3}}$ |
h. $2+\sqrt{3}$ |
3
We geven enkel de overige goniometrische getallen.
| a. $\cos\alpha = -\frac{2\sqrt{2}}{3} \quad,\quad \tan\alpha = -\frac{\sqrt{2}}{4}$ |
| b. $\sin\alpha = -\frac{\sqrt{15}}{4} \quad,\quad \tan\alpha = \sqrt{15}$ |
| c. $\cos\alpha = \frac{3}{5} \quad,\quad \sin\alpha = -\frac{4}{5}$ |
4
| a. Fout: de hoek ligt in het tweede kwadrant |
| b. Fout: $\frac{5\pi}{4}$ kan ook, net zoals elk geheel veelvoud van $2\pi$ bij deze hoeken. |
5
| a. $\sin\alpha = \frac{\sqrt{6}-\sqrt{2}}{4} \quad,\quad \cos\alpha = -\frac{\sqrt{2}+\sqrt{6}}{4}$ |
| b. $\frac{24}{25}$ |
| c. $0$ |
7
| a. $1$ |
b. $0$ |
c. $0$ |
d. $\frac{\cos(2\alpha)}{2}$ |
8
$\displaystyle \frac{10}{\sqrt{3}}$
9
$\displaystyle\frac{\sqrt{2}+\sqrt{3}}{2}$
10
Ja.
11
$\displaystyle \beta = \frac{\pi}{3} \quad,\quad A = \frac{8\sqrt{3}}{3}\quad,\quad C = \frac{4\sqrt{3}}{3}$
12
$\displaystyle 20\sqrt{3}$
13
$\displaystyle 2\sin\left(54^\circ\right) = \frac{\sin\left(108^\circ\right)}{\sin\left(36^\circ\right)}$
14
| a. $x = \frac{\pi}{6}+2k\pi \,\vee\, x=\frac{5\pi}{6}+2k\pi \quad (k \in \mathbb{Z})$ |
b. $x = \frac{-\pi}{8}+k\frac{\pi}{2} \quad (k \in \mathbb{Z})$ |
c. $x = \frac{\pi}{12}+k\frac{\pi}{2} \,\vee\, x=\frac{\pi}{3}+k\pi \quad (k \in \mathbb{Z})$ |
d. $x = \frac{\pi}{2}+k\pi \,\vee\, x=2k\pi \,\vee\, x=\frac{2\pi}{3}+2k\pi\,\vee\, x=\frac{4\pi}{3}+2k\pi \quad (k \in \mathbb{Z})$ |
15
| a. $\displaystyle \left\{\, -\frac{1}{2}\right\}$ |
b. $\displaystyle x= \sqrt{-\frac{1}{2}+\frac{\sqrt{13}}{6}} \vee x= -\sqrt{-\frac{1}{2}+\frac{\sqrt{13}}{6}}$ |
16
| a. $\displaystyle \left(2,\frac{\pi}{4}\right)$ |
c. $\displaystyle \left(\frac{1}{2},\frac{5\pi}{4}\right)$ |
e. $\displaystyle \left(5,\frac{3\pi}{2}\right)$ |
| b. $\displaystyle \left(3,\frac{\pi}{6}\right)$ |
d. $\displaystyle \left(2\sqrt{2},\frac{5\pi}{6}\right)$ |
f. $\displaystyle \left(2,\frac{2\pi}{3}\right)$ |
17
| a. $\displaystyle \left(-\frac{\sqrt{2}}{2},-\sqrt{\frac{3}{2}}\right)$ |
b. $\displaystyle \left(3,-\sqrt{3}\right)$ |
c. $\displaystyle \left(7,0\right)$ |
18
20
| a. $ z=2(\cos(\frac{\pi}{2})+i\cdot\sin(\frac{\pi}{2}))$ |
c. $z= \sqrt{13}(\cos(\theta)+i\cdot\sin(\theta))$ |
e. $z= cos(\frac{3\pi}{2})+i\cdot\sin(\frac{3\pi}{2})$ |
g. $z= cos(\frac{3\pi}{2})+i\cdot\sin(\frac{3\pi}{2})$ |
| b. $z=\sqrt{2}(cos(\frac{7\pi}{4})+i\cdot\sin(\frac{7\pi}{4}))$ |
d. $z= \sqrt{26}(\cos(\theta)+i\cdot\sin(\theta))$ |
f. $z= 10(\cos(\theta)+i\cdot\sin(\theta))$ |
h. $z= \frac{2\sqrt{5}}{5}(\cos(\theta)+i\cdot\sin(\theta))$ |
21
| a. $z_0=\cos(\frac{\pi}{4})+i\cdot\sin(\frac{\pi}{4}) \quad,\quad z_1=\cos(\frac{5\pi}{4})+i\cdot\sin(\frac{5\pi}{4})$ |
| b. $z_0=\cos(0)+i\cdot\sin(0) \quad,\quad z_1=\cos(\frac{2\pi}{3})+i\cdot\sin(\frac{2\pi}{3})
\quad,\quad z_2=\cos(\frac{4\pi}{3})+i\cdot\sin(\frac{4\pi}{3})$ |
| c. $z_0=2((\cos(\frac{\pi}{4})+i\cdot\sin(\frac{\pi}{4})) \quad,\quad z_1=2(\cos(\frac{3\pi}{4})+i\cdot\sin(\frac{3\pi}{4}))
\quad,\quad z_2=2(\cos(\frac{5\pi}{4})+i\cdot\sin(\frac{5\pi}{4})) \quad,\quad z_3=2(\cos(\frac{7\pi}{4})+i\cdot\sin(\frac{7\pi}{4}))$ |
| d. $z_0=2((\cos(0)+i\cdot\sin(0)) \quad,\quad z_1=2(\cos(\frac{\pi}{3})+i\cdot\sin(\frac{\pi}{3}))
\quad,\quad z_2=2(\cos(\frac{2\pi}{3})+i\cdot\sin(\frac{2\pi}{3})) \quad,\quad z_3=2(\cos(\pi)+i\cdot\sin(\pi))
\quad,\quad z_4=2(\cos(\frac{4\pi}{3})+i\cdot\sin(\frac{4\pi}{3})) \quad,\quad z_5=2(\cos(\frac{5\pi}{3})+i\cdot\sin(\frac{5\pi}{3}))$ |