if cos(t) = 1/3 and sin(t) < 0 find sin(t) and tan(t) using trigonometric identities.

If cos(t) = 1/3 and sin(t) is negative,

this is a picture of the triangle ∠t makes:

Now we can use the Pythagorean theorem to find sin(t).

sin²(t) + (1/3)² = 1²

sin²(t) = 1 - 1/9

sin(t) = \(\pm\sqrt{\frac89}=\pm\frac{\sqrt8}{3}\)

The prolem says that sin < 0 , so we know that it is the negative option.

sin(t) = \(-\frac{2\sqrt2}{3}\)

tan = sin / cos

tan(t) = sin(t) / cos(t)

tan(t) = \(-\frac{2\sqrt2}{3}/\frac13\)

tan(t) = \(-\frac{2\sqrt2}{3}\cdot\frac31\)

tan(t) = \(-2\sqrt2\)