Question

How do you use fundamental identities to find the values of the trigonometric values given sinx = -12/13 and cot > 0?

Answer 1

If cotangent of `x` is larger than zero, it's sister function, tangent, must be positive as well.

Refer to the following image to see the signs of the functions depending on the quadrant.

This means that cosine is positive in quadrant IV, all are positive in quadrant I, et cetera, et cetera.

So, we know that sine is negative and tangent is positive. Tangent is positive in quadrants `I` and `III`. However, if `x` was in quadrant `I`, sine would be positive. So, `x` is in quadrant `III`.

We can now use pythagorean theorem to determine the adjacent side to `x`, which will be needed to find the cos, sec, cot and tan ratios.

Let the unknown side be `y`.

`y^2 + (-12)^2 = 13^2`

`y^2 + 144 = 169`

`y^2 = 25`

`y = +-5`

Since cosine is negative in quadrant `3`, and the hypotenuse cannot be negative, `y = -5`.

We now apply the definitions of the ratios to determine the ratios.

`cosx = "adjacent"/"hypotenuse" = (-5)/13`

`secx = "hypotenuse"/"adjacent" = -13/5`

`cscx = "hypotenuse"/"opposite" = -13/12`

`tanx = "opposite"/"adjacent" = "12/5`

`cotx = "adjacent"/"opposite" = 5/12`

Hopefully this helps!