How do you find the exact value of the six trigonometric functions of the angle whose terminal side passes through $(x, 4x)$ ?

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Answer 1 · 2017-05-10T07:13:33

$sintheta=4/sqrt17$ , $costheta=1/sqrt17$ , $tantheta=4$ ,

$cottheta=1/4$ , $sectheta=sqrt17$ , $csctheta=sqrt17/4$

As the terminal side passes through $(x,4x)$ , we have $y=4x$ and hence its distance from origin is $sqrt(x^2+(4x)^2)=sqrt(x^2+16x^2)=xsqrt17$

Now consider the diagram below for a typical $theta$ , whose six trigonometrical ratios are

$sintheta=y/r$ , $costheta=x/r$ , $tantheta=y/x$ and

$cottheta=x/y$ , $sectheta=r/x$ , $csctheta=r/y$

enter image source here

As we have $y=4x$ and $r=xsqrt17$ ,

the six trigonometric ratios are

$sintheta=(4x)/(xsqrt17)=4/sqrt17$ , $costheta=(x)/(xsqrt17)=1/sqrt17$ ,

$tantheta=(4x)/x=4$ , $cottheta=x/(4x)=1/4$ ,

$sectheta=(xsqrt17)/x=sqrt17$ , $csctheta=(xsqrt17)/(4x)=sqrt17/4$

How do you find the exact value of the six trigonometric functions of the angle whose terminal side passes through (x, 4x)(x, 4x)?