If $csc theta=4/3$ , what is the sin, cos, tan, sec, and cot?

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Answer 1 · 2018-03-27T15:43:28

See below.

Instead of using formulas, it'd be easier to solve it geometrically, with a right triangle.

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Since $csc theta = 1/sintheta = "hypotenuse"/"opposite"=c/a = 4/3$ , this means that $a$ and $c$ are multiples of $3$ and $4$ , respectively.

In other words, we have $c=4k$ and $a=3k$ , for a real number $k$ .
By the Pythagorean theorem, $b = sqrt(c^2-a^2) = sqrt(16k^2-9k^2) = sqrt(7)*k$ .

Finally, for trigonometric functions :

$sin theta = "opposite"/"hypotenuse" = a/c = 3/4$
$cos theta = "adjacent"/"hypotenuse" = b/c = sqrt7/4$

$tan theta = "opposite"/"adjacent" = a/b = 3/sqrt7$
$cot theta = 1/tan theta = b/a = sqrt7/3$

$sec theta = "hypotenuse"/"adjacent" = c/b = 4/sqrt7$ .

Answer 2 · 2018-04-02T11:46:05

As below.

$csc theta = 4/3$

![https://hononegah.learning.powerschool.com/hhearn/2014-2015honorspre-calculus/cms_page/view/16304893]( useruploads.socratic.org )

$sin theta = 1/csc theta = 1 / (4/3) = 3/4$

$cos^2 theta = 1 - sin^2 theta = 1 - 9 / 16 = 7 / 16$

$cos theta = +- sqrt7 / 4$

$sec theta = 1 / cos theta = +- 4 / sqrt7$

$tan theta = sin theta / cos theta = +- (3/4) / (sqrt7 / 4) = +-3 / sqrt7$

$cot theta = 1 / tan theta = +- sqrt7 / 3$

If csc theta=4/3csc theta=4/3, what is the sin, cos, tan, sec, and cot?