How do you find the exact value of the six trigonometric functions of the angle whose terminal side passes through $(\frac{2}{3}, \frac{5}{2})$ $(2/3, 5/2)$ ?

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How do you find the exact value of the six trigonometric functions of the angle whose terminal side passes through $(\frac{2}{3}, \frac{5}{2})$ $(2/3, 5/2)$ ?

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Answer 1 · 2018-01-11T00:25:26

$cos (θ) = \frac{4}{\sqrt{241}}$ $cos(theta)=4/sqrt(241)$ , $sin (θ) = \frac{15}{\sqrt{241}}$ $sin(theta)=15/sqrt(241)$ , and $tan (θ) = \frac{15}{4}$ $tan(theta)=15/4$ , $sec (θ) = \frac{\sqrt{241}}{4}$ $sec(theta)=sqrt(241)/4$ , $csc (θ) = \frac{\sqrt{241}}{15}$ $csc(theta)=sqrt(241)/15$ , and $cot (θ) = \frac{4}{15}$ $cot(theta)=4/15$ .

Explanation:

The points $(\frac{2}{3}, \frac{5}{2})$ $(2/3,5/2)$ and $(4, 15)$ $(4,15)$ fall on the same line and therefore the terminal side of the same angle, $θ$ $theta$ , and the second point is easier to work with. (I basically multiplied through by 6 to scale the point up.)

Calculate $r = \sqrt{4^{2} + 15^{2}} = \sqrt{16 + 225} = \sqrt{241}$ $r=sqrt(4^2+15^2)=sqrt(16+225)=sqrt(241)$ .

Knowing $x$ $x$ , $y$ $y$ , and $r$ $r$ we can find:

$cos (θ) = \frac{x}{r}$ $cos(theta)=x/r$ , $sin (θ) = \frac{y}{r}$ $sin(theta)=y/r$ , and $tan (θ) = \frac{y}{x}$ $tan(theta)=y/x$ .

For this problem we have:

$cos (θ) = \frac{4}{\sqrt{241}}$ $cos(theta)=4/sqrt(241)$ , $sin (θ) = \frac{15}{\sqrt{241}}$ $sin(theta)=15/sqrt(241)$ , and $tan (θ) = \frac{15}{4}$ $tan(theta)=15/4$ .

The remaining functions are found by taking reciprocals:

$sec (θ) = \frac{\sqrt{241}}{4}$ $sec(theta)=sqrt(241)/4$ , $csc (θ) = \frac{\sqrt{241}}{15}$ $csc(theta)=sqrt(241)/15$ , and $cot (θ) = \frac{4}{15}$ $cot(theta)=4/15$ .

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How do you find the exact value of the six trigonometric functions of the angle whose terminal side passes through $(\frac{2}{3}, \frac{5}{2})$ $(2/3, 5/2)$ ?

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How do you find the exact value of the six trigonometric functions of the angle whose terminal side passes through (2/3, 5/2)(23,52)(2/3, 5/2)?

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How do you find the exact value of the six trigonometric functions of the angle whose terminal side passes through $(\frac{2}{3}, \frac{5}{2})$ $(2/3, 5/2)$ ?