diff options
Solved 2nd exercise
2 files changed, 10 insertions, 2 deletions
| -rw-r--r-- | Year 1/Mathematical analysis/Homework1.pdf | bin | 32721 -> 34958 bytes | |||
| -rw-r--r-- | Year 1/Mathematical analysis/Homework1.tex | 12 |
diff --git a/Year 1/Mathematical analysis/Homework1.pdf b/Year 1/Mathematical analysis/Homework1.pdf Binary files differindex fe7da74..90bbe9a 100644 --- a/Year 1/Mathematical analysis/Homework1.pdf +++ b/Year 1/Mathematical analysis/Homework1.pdf diff --git a/Year 1/Mathematical analysis/Homework1.tex b/Year 1/Mathematical analysis/Homework1.tex index 861365b..cb9c7cb 100644 --- a/Year 1/Mathematical analysis/Homework1.tex +++ b/Year 1/Mathematical analysis/Homework1.tex @@ -45,12 +45,20 @@ Докажете тъждеството \begin{equation*} - \arcsin{\left( 2x \sqrt{1-x^2} \right)} = \pi - 2 \arccos(x)\text{, където } x \in \left[ \frac{-\sqrt{2}}{2}, \frac{\sqrt{2}}{2} \right] + \arcsin \left( 2x \sqrt{1-x^2} \right) = \pi - 2 \arccos \left( x \right) \text{, където } x \in \left[ \frac{-\sqrt{2}}{2}, \frac{\sqrt{2}}{2} \right] \end{equation*} \paragraph{Решение:} -[TODO] +\begin{align*} + & \arcsin \left( 2x \sqrt{1-x^2} \right) = \alpha \iff \sin \left( \alpha \right) = 2x \sqrt{1-x^2} \qquad \arccos \left( x \right) = \beta \iff \cos \left( \beta \right) = x \\ + & \text{За } x = \frac{-\sqrt{2}}{2} \;,\quad \sin(\alpha) = \cancel{2} \cdot \frac{-\sqrt{2}}{\cancel{2}} \cdot \sqrt{1 - \left( \frac{-\sqrt{2}}{2} \right)^2} = -\sqrt{2} \cdot \sqrt{\frac{1}{2}} = -1 \;,\; \cos \left( \beta \right) = -\frac{\sqrt{2}}{2} \\ + & \text{Аналогично, за } x = \frac{\sqrt{2}}{2} \;,\quad \sin \left( \alpha \right) = 1 \;,\quad \cos \left( \beta \right) = \frac{\sqrt{2}}{2} \\ + & \implies \alpha \in \left[ -\frac{\pi}{2}, \frac{\pi}{2} \right] \;,\; \beta \in \left[ -\frac{\pi}{4}, \frac{\pi}{4} \right] \implies \sin \left( \beta \right) = +\sqrt{1 - \cos ^ 2 \left( \beta \right)} = \sqrt{1 - x ^ 2} \\ + & \omit\hfill \\ + & \sin \left( \alpha \right) = 2x\sqrt{1 - x^2} = 2\cos \left( \beta \right)\sin \left( \beta \right) = \sin \left( 2\beta \right) = \sin \left( \pi - 2\beta \right) \qquad\quad 2\beta \in \left[ -\frac{\pi}{2}, \frac{\pi}{2} \right] \\ + & \colorbox{yellow}{$\implies \alpha = \pi - 2\beta$} +\end{align*} \hfill |