Question

find tan θ for the angle shown: the terminal side is in quadrant iii at 225

Explanation:

Step1: Find the reference angle

The reference angle $\theta_{ref}$ for an angle $\theta = 225^{\circ}$ in Quadrant III is $\theta_{ref}=225^{\circ}- 180^{\circ}=45^{\circ}$.

Step2: Determine the sign of $\tan\theta$

In Quadrant III, both $\sin\theta$ and $\cos\theta$ are negative. Since $\tan\theta=\frac{\sin\theta}{\cos\theta}$, the quotient of two negative numbers is positive.

Step3: Calculate $\tan\theta$ using the reference angle

We know that $\tan45^{\circ} = 1$. Since $\tan\theta$ has the same value (in terms of magnitude) as $\tan\theta_{ref}$ for $\theta$ and $\theta_{ref}$ related as above and $\tan\theta>0$ in Quadrant III, $\tan225^{\circ}=\tan45^{\circ}$.

Answer:

$1$