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#