# If sin theta=4/5 and theta is in quadrant 2, what is the value of tan theta?

May 8, 2016

$\tan \theta = \frac{4}{- 3} \mathmr{and} - \frac{4}{3}$

#### Explanation:

$S \in \theta = \frac{4}{5}$ means that in the right-angled triangle, the opposite side is 4 and the hypotenuse is 5. the adjacent side must therefore be 3. (Pythagorean triple 3,4,5)

However, as this is an angle in the 2nd quadrant, the adjacent side is an $x$ value and must be -3.

$\tan \theta = \text{opposite"/"adjacent}$

This leads to $\tan \theta = \frac{4}{- 3} \mathmr{and} - \frac{4}{3}$

This agrees with what we know from the CAST rule that Tan values are negative in the second quadrant.

Once you know the lengths of the sides of the triangle, you can write down the value of any of the trig ratio's for the angle $\theta$

We have
$a \mathrm{dj} = x = 3$
$o p p = y = 4$
$h y p = r = 5$

The signs will change depending on what quadrant the triangle is in.