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

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

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Answer 1 · 2017-05-10T07:13:33

$sin θ = \frac{4}{\sqrt{17}}$ $sintheta=4/sqrt17$ , $cos θ = \frac{1}{\sqrt{17}}$ $costheta=1/sqrt17$ , $tan θ = 4$ $tantheta=4$ ,

$cot θ = \frac{1}{4}$ $cottheta=1/4$ , $sec θ = \sqrt{17}$ $sectheta=sqrt17$ , $csc θ = \frac{\sqrt{17}}{4}$ $csctheta=sqrt17/4$

Explanation:

As the terminal side passes through $(x, 4 x)$ $(x,4x)$ , we have $y = 4 x$ $y=4x$ and hence its distance from origin is $\sqrt{x^{2} + {(4 x)}^{2}} = \sqrt{x^{2} + 16 x^{2}} = x \sqrt{17}$ $sqrt(x^2+(4x)^2)=sqrt(x^2+16x^2)=xsqrt17$

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

$sin θ = \frac{y}{r}$ $sintheta=y/r$ , $cos θ = \frac{x}{r}$ $costheta=x/r$ , $tan θ = \frac{y}{x}$ $tantheta=y/x$ and

$cot θ = \frac{x}{y}$ $cottheta=x/y$ , $sec θ = \frac{r}{x}$ $sectheta=r/x$ , $csc θ = \frac{r}{y}$ $csctheta=r/y$

enter image source here

As we have $y = 4 x$ $y=4x$ and $r = x \sqrt{17}$ $r=xsqrt17$ ,

the six trigonometric ratios are

$sin θ = \frac{4 x}{x \sqrt{17}} = \frac{4}{\sqrt{17}}$ $sintheta=(4x)/(xsqrt17)=4/sqrt17$ , $cos θ = \frac{x}{x \sqrt{17}} = \frac{1}{\sqrt{17}}$ $costheta=(x)/(xsqrt17)=1/sqrt17$ ,

$tan θ = \frac{4 x}{x} = 4$ $tantheta=(4x)/x=4$ , $cot θ = \frac{x}{4 x} = \frac{1}{4}$ $cottheta=x/(4x)=1/4$ ,

$sec θ = \frac{x \sqrt{17}}{x} = \sqrt{17}$ $sectheta=(xsqrt17)/x=sqrt17$ , $csc θ = \frac{x \sqrt{17}}{4 x} = \frac{\sqrt{17}}{4}$ $csctheta=(xsqrt17)/(4x)=sqrt17/4$

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

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Explanation:

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

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Explanation:

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