Trigonometria analytica: introdvctio

Antequam vnitatem incipias, pvnctvm vnde proficisceris metire. Brevis est sui ipsivs probatio: non aestimatvr, et eivs exitvs in solo tvo navigatro servatvr.

1. Qvaestio 1Qvae ex aeqvalitatibvs infra positis identitas est in dominio svo commvni?

   $\sec x - \cos x = \sin x \tan x$

   $\sqrt{1 - \cos^2 x} = \sin x$

   $\sin x + \cos x = 1$

   $\frac{\tan x}{\sin x} = \cos x$

2. Qvaestio 2Si $\tan \theta = -\frac{3}{4}$ et $\theta$ in secvndo qvadrante iacet, compvta $\cos \theta$.

3. Qvaestio 3Compvta $\tan \frac{7\pi}{12}$, scribendo $\frac{7\pi}{12} = \frac{\pi}{3} + \frac{\pi}{4}$.

   $-2 - \sqrt{3}$

   $1 + \sqrt{3}$

   $2 + \sqrt{3}$

   $2 - \sqrt{3}$

4. Qvaestio 4Si $\sin \theta = \frac{3}{5}$ et $\frac{\pi}{2} < \theta < \pi$, compvta $\sin 2\theta$.

5. Qvaestio 5Compvta valorem maximvm fvnctionis $f(x) = 3 \sin x - 4 \cos x + 1$.

6. Qvaestio 6Qvot solvtiones habet aeqvatio $2 \cos^2 x - \cos x - 1 = 0$ in $\ico{0}{2\pi}$?

7. Qvaestio 7Qvi aeqvationem $\cos x - \sin x = 1$ ad qvadratvm elevando solvit candidatos $x = \frac{k\pi}{2}$ obtinet. Qvae svnt solvtiones in $\ico{0}{2\pi}$?

   $0$ et $\frac{3\pi}{2}$

   $0$, $\frac{\pi}{2}$, $\pi$ et $\frac{3\pi}{2}$

   $0$ et $\frac{\pi}{2}$

   $\frac{\pi}{2}$ et $\pi$

8. Qvaestio 8Qvem angvlvm efficivnt rectae $y = 2x + 1$ et $y = -3x + 4$?

   $\frac{\pi}{4}$

   $\frac{3\pi}{4}$

   $\arctan 5$

   $\frac{\pi}{2}$

9. Qvaestio 9Qvid repraesentat aeqvatio $x^2 + 4xy + y^2 - 3 = 0$?

   Hyperbola.

   Ellipsis.

   Circvmferentia.

   Parabola.

10. Qvaestio 10Ex trvnco cvivs sectio transversa circvlaris est, diametro $20$ centimetrorvm, trabs sectionis transversae rectangvlaris excidenda est, ita vt qvattvor vertices sectionis in circvmferentia trvnci iaceant. Si $\theta$ est angvlvs qvem diagonalis rectangvli cvm vno e lateribvs eivs efficit, qvanta est maxima area sectionis possibilis, in centimetris qvadratis?

11. Qvaestio 11Ex vnitatibvs prioribvsQvae svnt coordinatae pvncti $P\left( -\frac{7\pi}{6} \right)$ circvmferentiae vnitariae?

    $\left( -\frac{\sqrt{3}}{2}, \frac{1}{2} \right)$

    $\left( -\frac{\sqrt{3}}{2}, -\frac{1}{2} \right)$

    $\left( \frac{\sqrt{3}}{2}, \frac{1}{2} \right)$

    $\left( -\frac{1}{2}, \frac{\sqrt{3}}{2} \right)$

12. Qvaestio 12Ex vnitatibvs prioribvsCompvta $\arccos\left( \cos \frac{4\pi}{3} \right)$.

    $\frac{2\pi}{3}$

    $\frac{4\pi}{3}$

    $-\frac{2\pi}{3}$

    $\frac{\pi}{3}$

13. Qvaestio 13Ex vnitatibvs prioribvsCompvta cosinvm angvli qvem vectores $\left( 3, 4 \right)$ et $\left( 4, 3 \right)$ efficivnt.