The point (8,-15) is on the terminal side of an angle in standard position, how do you determine the exact values of the six trigonometric functions of the angle?

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Answer 1 · 2017-08-16T16:31:52

Please see below.

Memorize this

$(x, y)$ $(x,y)$ lies on the terminal side of $θ$ $theta$ , then

$r = \sqrt{x^{2} + y^{2}}$ $r = sqrt(x^2+y^2)$ and

$sin θ = \frac{y}{r}$ $sin theta = y/r$ $" "$ $" "$ $" "$ $csc θ = \frac{r}{y}$ $csc theta = r/y$

$cos θ = \frac{x}{r}$ $cos theta = x/r$ $" "$ $" "$ $" "$ $sec θ = \frac{r}{x}$ $sec theta = r/x$

$tan θ = \frac{y}{x}$ $tan theta = y/x$ $" "$ $" "$ $" "$ $cot θ = \frac{x}{y}$ $cot theta = x/y$

For this question

We have $(x, y) = (8, - 15)$ $(x,y) = (8,-15)$
Do the arithmetic and substitute.

$r = \sqrt{{(8)}^{2} + {(- 15)}^{2}} = 17$ $r = sqrt((8)^2+(-15)^2) = 17$

So

$sin θ = - \frac{15}{17}$ $sin theta = -15/17$ $" "$ $" "$ $" "$ $csc θ = - \frac{17}{15}$ $csc theta =-17/15$

$cos θ = \frac{8}{17}$ $cos theta = 8/17$ $" "$ $" "$ $" "$ $sec θ = \frac{17}{8}$ $sec theta = 17/8$

$tan θ = - \frac{15}{8}$ $tan theta = -15/8$ $" "$ $" "$ $" "$ $cot θ = - \frac{8}{15}$ $cot theta = -8/15$