Question

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

Answer 1

See below.

Explanation:

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

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

As below.

Explanation:

`csc theta = 4/3`

`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`