# If sec displaystylealpha=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}}

Question
If sec $$\displaystyle\alpha=\frac{41}{{9}},{0}<\alpha<\frac{\pi}{{2}},$$</span> then find the exact value of each of the following.
a) $$\displaystyle{\sin},\frac{\alpha}{{2}}$$
b) $$\displaystyle{\cos},\frac{\alpha}{{2}}$$
c) $$\displaystyle{\tan},\frac{\alpha}{{2}}$$

2020-11-08
Step 1
Given that,
$$\displaystyle \sec{\alpha}=\frac{41}{{9}}\ \text{and}\ \alpha$$ is in furst quadrant.
In first quandrant, all the trigonometric angles are positive.
$$\displaystyle \sec{\alpha}=\frac{{{h}{y}{p}}}{{{a}{d}{j}}}=\frac{41}{{9}}$$
Using Pythagorean Theorem,
$$\displaystyle{\left({o}{p}{p}\right)}^{2}={\left({h}{y}{p}\right)}^{2}-{\left({a}{d}{j}\right)}^{2}$$
$$\displaystyle={\left({41}\right)}^{2}-{\left({9}\right)}^{2}$$
$$= 1681\ -\ 81$$
$$= 1600$$
$$\displaystyle{o}{p}{p}=\pm\sqrt{{1600}}$$
$$\displaystyle{o}{p}{p}=\pm{40}$$
Since, all the angles area positive we take $$opp = 40$$
Step 2
a) $$\displaystyle{\sin},\frac{\alpha}{{2}}:$$
Using the formula,
$$\displaystyle{\sin},\frac{\alpha}{{2}}=\frac{\sqrt{{{1}- \cos{\alpha}}}}{{2}}$$ ... (1)
$$\displaystyle \cos{\alpha}=\frac{1}{{ \sec{\alpha}}}=\frac{1}{{\frac{41}{{9}}}}=\frac{9}{{41}}$$
Step 3
Substitute the above value in equation (1), we get
$$\displaystyle{\sin},\frac{\alpha}{{2}}=\sqrt{{\frac{{{1}- \cos{\alpha}}}{{2}}}}$$
$$\displaystyle{\sin},\frac{\alpha}{{2}}=\sqrt{{\frac{{{1}-\frac{9}{{41}}}}{{2}}}}$$
$$\displaystyle=\sqrt{{\frac{{\frac{{{41}-{9}}}{{41}}}}{{2}}}}$$
$$\displaystyle=\sqrt{{\frac{32}{{{41}\times{2}}}}}$$
$$\displaystyle=\sqrt{{\frac{16}{{41}}}}$$
$$\displaystyle=\frac{\sqrt{{16}}}{\sqrt{{41}}}$$
$$\displaystyle=\frac{4}{\sqrt{{41}}}$$
$$\displaystyle=\frac{4}{\sqrt{{41}}}\times\frac{\sqrt{{41}}}{\sqrt{{41}}}$$
$$\displaystyle{\sin},\frac{\alpha}{{2}}=\frac{{{4}\sqrt{{41}}}}{{41}}$$
Step 4
b) $$\displaystyle{\cos},\frac{\alpha}{{2}}$$
Using the formula,
$$\displaystyle{\cos},\frac{\alpha}{{2}}=\sqrt{{\frac{{{1}+ \cos{\alpha}}}{{2}}}}$$..(2)
Substitute $$\displaystyle \cos{\alpha}=\frac{9}{{41}}$$ in the above formula, we get
$$\displaystyle{\cos},\frac{\alpha}{{2}}=\sqrt{{\frac{{{1}+ \cos{\alpha}}}{{2}}}}$$
$$\displaystyle=\sqrt{{\frac{{{1}+\frac{9}{{41}}}}{{2}}}}$$
$$\displaystyle=\sqrt{{\frac{50}{{{41}\times{2}}}}}$$
$$\displaystyle=\sqrt{{\frac{25}{{41}}}}$$
$$\displaystyle=\frac{\sqrt{{25}}}{\sqrt{{41}}}$$
$$\displaystyle=\frac{5}{\sqrt{{41}}}$$
$$\displaystyle=\frac{5}{\sqrt{{41}}}\times\frac{\sqrt{{41}}}{\sqrt{{41}}}$$
$$\displaystyle{\cos},\frac{\alpha}{{2}}=\frac{{{5}\sqrt{{41}}}}{{41}}$$
Step 5
c) $$\displaystyle{\tan},\frac{\alpha}{{2}}$$
Using the formula,
$$\displaystyle{\tan},\frac{\alpha}{{2}}=\frac{{{1}- \cos{\alpha}}}{{ \sin{\alpha}}}$$...(3)
$$\displaystyle \sin{\alpha}=\frac{{{o}{p}{p}}}{{{h}{y}{p}}}=\frac{40}{{41}}$$
$$\displaystyle \cos{\alpha}=\frac{9}{{41}}$$
Substitute the above values in equation (3), we get
$$\displaystyle{\tan},\frac{\alpha}{{2}}=\frac{{{1}- \cos{\alpha}}}{{ \sin{\alpha}}}$$
$$\displaystyle=\frac{{{1}-\frac{9}{{41}}}}{{\frac{40}{{41}}}}$$
$$\displaystyle=\frac{{\frac{32}{{41}}}}{{\frac{40}{{41}}}}$$
$$\displaystyle=\frac{32}{{40}}$$
$$\displaystyle{\tan},\frac{\alpha}{{2}}=\frac{4}{{5}}$$

### Relevant Questions

Find the exact value of each of the remaining trigonometric function of $$\theta.$$
$$\displaystyle \cos{\theta}=\frac{24}{{25}},{270}^{\circ}<\theta<{360}^{\circ}$$
$$\displaystyle \sin{\theta}=?$$
$$\displaystyle \tan{\theta}=?$$
$$\displaystyle \sec{\theta}=?$$
$$\displaystyle \csc{\theta}=?$$
$$\displaystyle \cot{\theta}=?$$
Find and solve the exact value of each of the following under the given conditions $$\tan\ \alpha =\ -\frac{7}{24},\ \alpha$$ lies in quadrant 2,
$$\cos\ \beta = \frac{3}{4},\ \beta$$ lies in quadrant 1
a. $$\sin(\alpha\ +\ \beta)$$
b. $$\cos(\alpha\ +\ \beta)$$
c. $$\tan (\alpha\ +\ \beta)$$
Given $$\displaystyle \sin{{\left(\alpha\right)}}=\frac{4}{{9}}{\quad\text{and}\quad}\pi\text{/}{2}<\alpha<\pi$$,
find the exact value of $$\displaystyle \sin{{\left(\alpha\text{/}{2}\right)}}.$$
Use properties of the Laplace transform to answer the following
(a) If $$f(t)=(t+5)^2+t^2e^{5t}$$, find the Laplace transform,$$L[f(t)] = F(s)$$.
(b) If $$f(t) = 2e^{-t}\cos(3t+\frac{\pi}{4})$$, find the Laplace transform, $$L[f(t)] = F(s)$$. HINT:
$$\cos(\alpha + \beta) = \cos(\alpha)\cos(\beta) - \sin(\alpha) \sin(\beta)$$
(c) If $$F(s) = \frac{7s^2-37s+64}{s(s^2-8s+16)}$$ find the inverse Laplace transform, $$L^{-1}|F(s)| = f(t)$$
(d) If $$F(s) = e^{-7s}(\frac{1}{s}+\frac{s}{s^2+1})$$ , find the inverse Laplace transform, $$L^{-1}[F(s)] = f(t)$$
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.
Use a right triangle to write the following expression as an algebraic expression. Assume that x is positive and in the domain of the given inverse trigonometric function.
Given:
$$\displaystyle \tan{{\left({{\cos}^{ -{{1}}}{5}}{x}\right)}}=?$$
A 1000.0 kg car is moving at 15 km/h If a 2000.0 kg truck has 23 times the kinetic energy of the car, how fast is thetruck moving?
a. 51 km/h
b. 41 km/h
c. 61 km/h
d. 72 km/h
a) Find the rational zeros and then the other zeros of the polynomial function $$\displaystyle{\left({x}\right)}={x}^{3}-{4}{x}^{2}+{2}{x}+{4},\ \tet{that is, solve}\ \displaystyle f{{\left({x}\right)}}={0}.$$
b) Factor $$f(x)$$ into linear factors.
The bulk density of soil is defined as the mass of dry solidsper unit bulk volume. A high bulk density implies a compact soilwith few pores. Bulk density is an important factor in influencing root development, seedling emergence, and aeration. Let X denotethe bulk density of Pima clay loam. Studies show that X is normally distributed with $$\displaystyle\mu={1.5}$$ and $$\displaystyle\sigma={0.2}\frac{{g}}{{c}}{m}^{{3}}$$.
(b) Find the probability that arandomly selected sample of Pima clay loam will have bulk densityless than $$\displaystyle{0.9}\frac{{g}}{{c}}{m}^{{3}}$$.
(c) Would you be surprised if a randomly selected sample of this type of soil has a bulkdensity in excess of $$\displaystyle{2.0}\frac{{g}}{{c}}{m}^{{3}}$$? Explain, based on theprobability of this occurring.
(a) $$\displaystyle{27}^{{{1}\text{/}{3}}}$$
(b) $$\displaystyle{\left(-{8}\right)}^{{{1}\text{/}{3}}}$$
(c) $$\displaystyle-{\left(\frac{1}{{8}}\right)}^{{{1}\text{/}{3}}}$$