# Let x=asin theta in sqrt{a^{2}-x^{2}}. Then find cos theta tan theta

Question
Trigonometric Functions
Let $$x=a\sin \theta in \sqrt{a^{2}-x^{2}}$$. Then find $$\cos \theta \ \tan \theta$$

2021-01-25
$$\sqrt{a^{2}-s^{2}\sin^{2}(\theta)}=$$
Factor out common term $$a^{2}$$
$$=\sqrt{a^{2}(1-\sin^{2}(\theta))}$$
Apply radical rule $$\sqrt[n]{ab}=\sqrt[n]{a}\sqrt[n]{b}$$, assuming $$a\geq0 / b\geq0$$
$$=\sqrt{a^{2}}\sqrt{-\sin^{2}(\theta)+1}$$
Apply radical rule $$\sqrt[n]{a^{n}}=a$$, assuming $$a\geq0$$ Use the following identity: $$\cos^{2}(x)+\sin^{2}(x)=1$$
Therefore $$1-\sin^{2}(x)=\cos^{2}(x)$$
$$=a\sqrt{\cos^{2}(\theta)}=a\cos(\theta)$$
Therefore $$\sqrt(a^{2}-a^{2}\sin^{2}(\theta))=a\cos(\theta)$$
$$\implies \cos \theta=\sqrt{a^{2}-a^{2}\frac{\sin^{2}(\theta)}}{a}$$
$$\implies\frac{a\sin(\theta)}{\sqrt{a^{2}-a^{2}\sin^{2}(\theta)}}=\tan(\theta)$$
$$\sqrt{a^{2}-a^{2}\sin^{2}(\theta)}=a\cos(\theta)$$

### Relevant Questions

Prove:
a) that x = Asin wt is a solution of theequation of motion:
$$\displaystyle{m}\dot{{\frac{{{d}^{{{2}}}{x}}}{{{\left.{d}{t}\right.}^{{{2}}}}}}}=-{k}\dot{{x}}$$ of the harmonic oscillator.
b) that $$\displaystyle{w}=\sqrt{{{k}}}{\left\lbrace{m}\right\rbrace}$$
c) that the momentum is given by p = $$\displaystyle{m}{w}{A}{\cos{{w}}}{t}$$
d) that the particle is stationary when x = A
(Hint: Use x = A sinwt to find the time when x =A , and then use this time in an expression for thevelocity.)
e) that the energy of a harmonic oscillator is $$\displaystyle{\frac{{{1}}}{{{2}}}}{k}{A}^{{{2}}}$$
To find: The equivalent polar equation for the given rectangular-coordinate equation.
Given:
$$\displaystyle\ {x}=\ {r}{\cos{\theta}}$$
$$\displaystyle\ {y}=\ {r}{\sin{\theta}}$$
b. From rectangular to polar:
$$\displaystyle{r}=\pm\sqrt{{{x}^{{{2}}}\ +\ {y}^{{{2}}}}}$$
$$\displaystyle{\cos{\theta}}={\frac{{{x}}}{{{r}}}},{\sin{\theta}}={\frac{{{y}}}{{{r}}}},{\tan{\theta}}={\frac{{{x}}}{{{y}}}}$$
Calculation:
Given: equation in rectangular-coordinate is $$\displaystyle{y}={x}$$.
Converting into equivalent polar equation -
$$\displaystyle{y}={x}$$
Put $$\displaystyle{x}={r}{\cos{\theta}},\ {y}={r}{\sin{\theta}},$$
$$\displaystyle\Rightarrow\ {r}{\sin{\theta}}={r}{\cos{\theta}}$$
$$\displaystyle\Rightarrow\ {\frac{{{\sin{\theta}}}}{{{\cos{\theta}}}}}={1}$$
$$\displaystyle\Rightarrow\ {\tan{\theta}}={1}$$
Thus, desired equivalent polar equation would be $$\displaystyle\theta={1}$$
Sketch a right triangle corresponding to the trigonometric function of the acute angle theta. Use the Pythagorean Theorem to determine the third side and then find the other five trigonometric functions of theta. $$\displaystyle{\cos{\theta}}=\frac{{21}}{{5}}$$
Let theta be an angle in standard position, $$\displaystyle{\sin{\theta}}{<}{0},{\cos{\theta}}{>}{0}$$. Name the quadrant in which theta lies.

sec theta = -3, tan theta > 0ZSK. Find the exact value of the remaining trigonometric functions of
thetaZSK.
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) $$\displaystyle{\tan},\frac{\alpha}{{2}}$$
comprehension check for derivatives of trigonometric functions:
a) true or false: if $$\displaystyle{f}{p}{r}{i}{m}{e}{\left(\theta\right)}=-{\sin{{\left(\theta\right)}}},{t}{h}{e}{n}{f{{\left(\theta\right)}}}={\cos{{\left(\theta\right)}}}.$$
b) true or false: $$\displaystyle{I}{f}\theta$$ is one of the non right angles in a right triangle and ЗІЛsin(theta) = 2/3ZSK, then the hypotenuse of the triangle must have length 3.
Which of the following are linear transformations from $$RR^{2} \rightarrow RR^{2} ?$$
(d) Rotation: if $$x = r \cos \theta, y = r \sin \theta,$$ then
$$\overrightarrow{T}(x,y)=(r \cos(\theta+ \varphi), r \sin (\theta+ \varphi))$$
for some constants $$\angle \varphi$$
(f) Reflection: given a fixed vector $$\overrightarrow{r} = (a, b), \overrightarrow{T}$$ maps each point to its reflection with
respect to $$\overrightarrow{r} \overrightarrow{T}(\overrightarrow{x})=\overrightarrow{x}-2\overrightarrow{x}_{r \perp}$$
$$=2 \overrightarrow{x}_{r}-\overrightarrow{x}$$
Use polar coordinates to find the limit. [Hint: Let $$\displaystyle{x}={r}{\cos{{\quad\text{and}\quad}}}{y}={r}{\sin{}}$$ , and note that (x, y) (0, 0) implies r 0.] $$\displaystyle\lim_{{{\left({x},{y}\right)}\to{\left({0},{0}\right)}}}\frac{{{x}^{{2}}-{y}^{{2}}}}{\sqrt{{{x}^{{2}}+{y}^{{2}}}}}$$
Find the exact value of each of the remaining trigonometric function of $$\theta.$$
$$\displaystyle \cos{\theta}=\frac{24}{{25}},{270}^{\circ}<\theta<{360}^{\circ}$$
$$\displaystyle \sin{\theta}=?$$
$$\displaystyle \tan{\theta}=?$$
$$\displaystyle \sec{\theta}=?$$
$$\displaystyle \csc{\theta}=?$$
$$\displaystyle \cot{\theta}=?$$