ok782 said:
Find the exact values for the following without using a calculator:
1 - Sin(Arcsin2)
2 - Arc cos(cos 7pi/4)
3 - csc(arc cos 2/9)
4 - tan(arc sin 3/7)Solve the following trigonometric equation for the domain 0 <= x <2pi
1- 2sin^2 x=0Thank you so much to anyone who has the chance to look this over. The test is tomorrow, so hopefully someone will see this today!
Some hints for the first lot here:
$\displaystyle \begin{align*} -1 \leq \sin{(x)} \leq 1 , \, -1 \leq \cos{(x)} \leq 1 \end{align*}$
The inverse trigonometric functions are defined over the following regions:
$\displaystyle \begin{align*} -\frac{\pi}{2} \leq \arcsin{(x)} \leq \frac{\pi}{2}, \, 0 \leq \arccos{(x)} \leq \pi , \, -\frac{\pi}{2} \leq \tan{(x)} \leq \frac{\pi}{2} \end{align*}$
I would be inclined to think that the first does not exist, as $\displaystyle \begin{align*} \arcsin{(2)} \end{align*}$ is saying there is an angle such that $\displaystyle \begin{align*} \sin{\left( \theta \right) } = 2 \end{align*}$, which is impossible.
2.
$\displaystyle \begin{align*} \cos{ \left( \frac{7\,\pi}{4} \right) } &= \cos{ \left( 2\,\pi - \frac{\pi}{4} \right) } \\ &= \cos{ \left( \frac{\pi}{4} \right) } \end{align*}$
and since $\displaystyle \begin{align*} 0 \leq \arccos{ (x) } \leq \pi \end{align*}$, that would have to mean
$\displaystyle \begin{align*} \arccos{ \left[ \cos{ \left( \frac{7\,\pi}{4} \right) } \right] } &= \arccos{ \left[ \cos{ \left( \frac{\pi}{4} \right) } \right] } \\ &= \frac{\pi}{4} \end{align*}$
3.
$\displaystyle \begin{align*} \arccos{ \left( \frac{2}{9} \right) } \end{align*}$ is in the first quadrant (can you see why?)
Drawing up a right angle triangle with an angle $\displaystyle \begin{align*} \theta \end{align*}$ having adjacent side 2 units and hypotenuse 9 units, by Pythagoras the opposite side must be $\displaystyle \begin{align*} \sqrt{77} \end{align*}$ units, therefore $\displaystyle \begin{align*} \arccos{ \left( \frac{2}{9} \right) } = \arcsin{ \left( \frac{\sqrt{77}}{9} \right) } \end{align*}$. So
$\displaystyle \begin{align*} \csc{ \left[ \arccos{ \left( \frac{2}{9} \right) } \right] } &= \csc{ \left[ \arcsin{ \left( \frac{\sqrt{77}}{9} \right) } \right] } \\ &= \frac{1}{\sin{ \left[ \arcsin{ \left( \frac{\sqrt{77}}{9} \right) } \right] }} \\ &= \frac{1}{\frac{\sqrt{77}}{9}} \\ &= \frac{9}{\sqrt{77}} \\ &= \frac{9\,\sqrt{77}}{77} \end{align*}$
Do you think your daughter could follow a similar process for question 4?
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ok782 said:
$\displaystyle \begin{align*} 1 - 2\sin^2{(x)} &= 0 \\ 2\sin^2{(x)} &= 1 \\ \sin^2{(x)} &= \frac{1}{2} \\ \sin{(x)} &= \pm \frac{1}{\sqrt{2}} \\ x &= \left\{ \frac{\pi}{4} , \, \frac{3\,\pi}{4} , \, \frac{5\,\pi}{4} , \, \frac{7\,\pi}{4} \right\} + 2\,\pi\,n \textrm{ where }n \in \mathbf{Z} \\ x &= \left\{ \frac{\pi}{4}, \, \frac{3\,\pi}{4}, \, \frac{5\,\pi}{4} , \, \frac{7\,\pi}{4} \right\} \textrm{ as } 0 \leq x \leq 2\,\pi \end{align*}$