Question 705435
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First reduce the fraction: *[tex \LARGE \frac{40}{30}\ =\ \frac{4}{3}]


Then use the fact that *[tex \LARGE \tan\varphi\ =\ \frac{\sin\varphi}{\cos\varphi}]:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \ \frac{\sin\varphi}{\cos\varphi}\ =\ \frac{4}{3}]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \ 4\cos\varphi\ =\ 3\sin\varphi]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \ 16\cos^2\varphi\ =\ 9\sin^2\varphi]


Use the Pythagorean Identity:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \ 16\cos^2\varphi\ =\ 9\ -\ 9\cos^2\varphi]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \ 25\cos^2\varphi\ =\ 9]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \ \cos^2\varphi\ =\ \frac{9}{25}]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \ \cos\varphi\ =\ \pm\frac{3}{5}\ =\ \pm 0.6000]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \ \sin^2\varphi\ =\ 1\ -\ \cos^2\varphi\ =\ 1\ =\ \frac{9}{25}\ =\ \frac{16}{25}]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \ \sin\varphi\ =\ \pm\frac{4}{5}\ =\ \pm 0.8000]


Note:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \ \tan\theta\ \geq\ 0\ \Right\ \text{sgn}(\sin\theta)\ =\ \text{sgn}(\cos\theta)]


Then


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \ \tan\varphi\ =\ \frac{4}{3}\ =\ 1.\overline{6666}]


For the other three functions, *[tex \LARGE \cot], *[tex \LARGE \sec], and *[tex \LARGE \csc] are the reciprocals of *[tex \LARGE \tan], *[tex \LARGE \cos], and *[tex \LARGE \sin] respectively.


John
*[tex \LARGE e^{i\pi}\ +\ 1\ =\ 0]
<font face="Math1" size="+2">Egw to Beta kai to Sigma</font>
My calculator said it, I believe it, that settles it
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