#### Didn’t find what you are looking for?

Question
Decimals
Given $$\displaystyle \sin{{\left(\alpha\right)}}=\frac{4}{{9}}{\quad\text{and}\quad}\pi\text{/}{2}<\alpha<\pi$$</span>,
find the exact value of $$\displaystyle \sin{{\left(\alpha\text{/}{2}\right)}}.$$

2021-03-10
Step 1
$$\displaystyle\frac{\pi}{{2}}<\alpha<\pi$$</span>
$$\displaystyle\frac{{\frac{\pi}{{2}}}}{{2}}<\frac{\alpha}{{2}}<\frac{\pi}{{2}}$$</span>
$$\displaystyle\frac{\pi}{{4}}<\frac{\alpha}{{2}}<\frac{\pi}{{2}}$$</span>
So, $$\displaystyle\frac{\alpha}{{2}}$$ in forst quadrant.
In first quadrant sin is positive.
Step 2 $$\displaystyle \sin{{\left(\frac{\alpha}{{2}}\right)}}=\sqrt{{\frac{{{1}- \cos{\alpha}}}{{2}}}}$$
So, we have to find cos alpha from $$\displaystyle \sin{\alpha}$$
$$\displaystyle\frac{\pi}{{2}}<\alpha<\pi$$</span> is second quadrant.
There $$\displaystyle \cos{\alpha}$$ is negative
$$\displaystyle \cos{\alpha}=-\sqrt{{{1}-{{\sin}^{2}\alpha}}}=-\sqrt{{{1}-{\left(\frac{4}{{9}}\right)}^{2}}}=-\sqrt{{{1}-\frac{16}{{81}}}}=-\frac{\sqrt{{65}}}{{9}}$$
Step 3
Plug $$\displaystyle \cos{\alpha}=-\frac{\sqrt{{65}}}{{9}}$$ in the formula
$$\displaystyle \sin{{\left(\frac{\alpha}{{2}}\right)}}=\sqrt{{\frac{{{1}- \cos{\alpha}}}{{2}}}}$$
$$\displaystyle \sin{{\left(\frac{\alpha}{{2}}\right)}}$$
$$\displaystyle=\sqrt{{\frac{{{1}-{\left(-\frac{\sqrt{{65}}}{{9}}\right)}}}{{2}}}}$$
$$\displaystyle=\sqrt{{\frac{{{1}+\frac{\sqrt{{65}}}{{9}}}}{{2}}}}$$
$$\displaystyle=\sqrt{{\frac{{{9}+\sqrt{{65}}}}{{18}}}}$$
Multiply numerator and denominator by 2
$$\displaystyle=\sqrt{{\frac{{{2}{\left({9}+\sqrt{{65}}\right)}}}{{{18}{\left({2}\right)}}}}}$$
$$\displaystyle=\sqrt{{\frac{{{18}+{2}\sqrt{{65}}}}{{36}}}}$$
$$\displaystyle=\sqrt{{{13}+{2}\sqrt{{13}}\frac{\sqrt{{{5}+{5}}}}{{6}}}}$$
$$\displaystyle=\frac{\sqrt{{{\left(\sqrt{{13}}+\sqrt{{5}}\right)}^{2}}}}{{6}}$$
$$\displaystyle=\frac{{\sqrt{{13}}+\sqrt{{5}}}}{{6}}$$
Answer: $$\displaystyle=\frac{{\sqrt{{13}}+\sqrt{{5}}}}{{6}}$$

### Relevant Questions

Given $$\displaystyle \csc{{\left({t}\right)}}={\left[\frac{{-{12}}}{{{7}}}\right]}$$\ \text{and}\ \displaystyle{\left[{\left(-\frac{\pi}{{2}}\right)}<{t}<{\left(\frac{\pi}{{2}}\right)}\right]}\) .
Find $$\sin\ t,\ \cos\ t\ \text{and}\ \tan\ t.$$ Give exact answers without decimals.
A true statement by inserting a symbol $$\displaystyle<,>{\quad\text{or}\quad}=$$ between the given numbers $$\displaystyle{0.58}\overline{{3}}{\quad\text{and}\quad}\frac{6}{{11}}$$
If sec $$\displaystyle\alpha=\frac{41}{{9}},{0}<\alpha<\frac{\pi}{{2}},$$ then find the exact value of each of the following.
a) $$\displaystyle{\sin},\frac{\alpha}{{2}}$$
b) $$\displaystyle{\cos},\frac{\alpha}{{2}}$$
c) $$\displaystyle{\tan},\frac{\alpha}{{2}}$$
Find the solution to this equation:
$$\displaystyle\sqrt{{2}} \cos{{\left({x}\right)}} \sin{{\left({x}\right)}}+ \cos{{\left({x}\right)}}={0}$$
The solution should be such that all angles are in radian. for solution the first angle should be between $$\displaystyle{\left[{0},{2}\pi\right)}$$ and then the period.
And when 2 or more solutions are available then the solution must be in increasing order of the angles.
The function $$\displaystyle{\left({9}{h}\right)}={8}{e}^{{-{0.4}{h}}}$$ can be used to determine the milligrams D of a certain drug in a patient's bloodstream h hours after the drug has been given. How many milligrams (to two decimals) will be resent in 7 years?
A true statement by inserting a symbol $$\displaystyle<,>{\quad\text{or}\quad}=$$ between the given numbers 7.123 and $$\displaystyle\frac{456}{{64}}.$$
$$\displaystyle{2}{\left({x}-{3}\right)}-{5}\le{3}{\left({x}+{2}\right)}-{18}$$
To make: A true statement by inserting a symbol $$<,\ >\ or\ =\ \text{between the given numbers}\ \displaystyle\overline{{0.6}}{\quad\text{and}\quad}\frac{5}{{6}}.$$
Equation: 8 divided by $$\displaystyle{\left({x}^{2}+{x}+{1}\right)}={1}$$
$$\displaystyle{x}={\left({s}{m}{a}{l}\le{r}{v}{a}{l}{u}{e}\right)}{x}={\left({l}{a}{r}\ge{r}{v}{a}{l}{u}{e}\right)}$$
The following quadratic function in general form, $$\displaystyle{S}{\left({t}\right)}={5.8}{t}^{2}—{81.2}{t}+{1200}$$ models the number of luxury home sales, S(t), in a major Canadian urban area, according to statistical data gathered over a 12 year period. Luxury home sales are defined in this market as sales of properties worth over \$3 Million (inflation adjusted). In this case, $$\displaystyle{\left\lbrace{t}\right\rbrace}={\left\lbrace{0}\right\rbrace}{Z}{S}{K}\ \text{represents}\ {2000}{\quad\text{and}\quad}{\left\lbrace{t}\right\rbrace}={\left\lbrace{11}\right\rbrace}$$represents 2011. Use a calculator to find the year when the smallest number of luxury home sales occurred. Without sketching the function, interpret the meaning of this function, on the given practical domain, in one well-expressed sentence.