Slovné úlohy s kosínusovou vetou
Teraz vyrieš tri slovné úlohy s použitím kosínusovej vety.
V akom zornom uhle sa Jakubovi ukazuje cesta dlhá 78 metrov, ak je od jedného jej konca vzdialený 56 metrov a od druhého konca 80 metrov?
\( a^{2} = b^{2} + c^{2} − 2bc\ \textrm{sin}\ α \)
\( a^{2} + 2bc\ \textrm{sin}\ α = b^{2} + c^{2} \)
\( 2bc\ \textrm{sin}\ α = b^{2} + c^{2} − a^{2} \)
\( \textrm{sin}\ α = \frac{b^{2}\ +\ c^{2}\ −\ a^{2}}{2bc} \)
\( α = \textrm{sin}^{−1} \left (\frac{b^{2}\ +\ c^{2}\ −\ a^{2}}{2bc} \right) \)
\( α = \textrm{sin}^{−1} \left (\frac{56^{2}\ +\ 80^{2}\ −\ 78^{2}}{2\ ·\ 56\ ·\ 80} \right) \)
\( α \doteq 30,00° \)
Není zaškrtnuto
\( a^{2} = b^{2} + c^{2} − 2bc\ \textrm{cos}\ α \)
\( a^{2} + 2bc\ \textrm{cos}\ α = b^{2} + c^{2} \)
\( 2bc\ \textrm{cos}\ α = b^{2} + c^{2} − a^{2} \)
\( \textrm{cos}\ α = \frac{b^{2}\ +\ c^{2}\ −\ a^{2}}{2bc} \)
\( α = \textrm{cos}^{−1} \left (\frac{b^{2}\ +\ c^{2}\ −\ a^{2}}{2bc} \right) \)
\( α = \textrm{cos}^{−1} \left (\frac{56^{2}\ +\ 80^{2}\ −\ 78^{2}}{2\ ·\ 56\ ·\ 80} \right) \)
\( α \doteq 67,34° \)
Není zaškrtnuto
\( a^{2} = b^{2} + c^{2} + 2bc\ \textrm{cos}\ α \)
\( a^{2} - 2bc\ \textrm{cos}\ α = b^{2} + c^{2} \)
\( 2bc\ \textrm{cos}\ α = a^{2} − b^{2} − c^{2} \)
\( \textrm{cos}\ α = \frac{a^{2}\ −\ b^{2}\ −\ c^{2}}{2bc} \)
\( α = \textrm{cos}^{−1} \left (\frac{a^{2}\ −\ b^{2}\ −\ c^{2}}{2bc} \right) \)
\( α = \textrm{cos}^{−1} \left (\frac{56^{2}\ −\ 80^{2}\ +\ 78^{2}}{2\ ·\ 56\ ·\ 80} \right) \)
\( α \doteq 45,00° \)
Není zaškrtnuto
\( a^{2} = b^{2} + c^{2} − 2bc\ \textrm{cos}\ β \)
\( a^{2} + 2bc\ \textrm{cos}\ β = b^{2} + c^{2} \)
\( 2bc\ \textrm{cos}\ β = b^{2} + c^{2} − a^{2} \)
\( \textrm{cos}\ β = \frac{b^{2}\ +\ c^{2}\ −\ a^{2}}{2bc} \)
\( β = \textrm{cos}^{−1} \left (\frac{b^{2}\ +\ c^{2}\ −\ a^{2}}{2bc} \right) \)
\( β = \textrm{cos}^{−1} \left (\frac{56^{2}\ +\ 80^{2}\ −\ 78^{2}}{2\ ·\ 56\ ·\ 80} \right) \)
\( β \doteq 67,34° \)
Není zaškrtnuto