Q.Determine the sign of $\cos200^\circ$ and $\tan320^\circ$, without using a calculator.

Each angle's sign can be read off directly once the quadrant it falls in is identified and the All–Sin–Tan–Cos rule is applied.

$200^\circ$ lies between $180^\circ$ and $270^\circ$ (QIII), where cosine is negative. $320^\circ$ lies between $270^\circ$ and $360^\circ$ (QIV), where tangent is negative.

Step 1 — locate $200^\circ$. Since $180^\circ<200^\circ<270^\circ$, this angle is in the third quadrant. By the All–Sin–Tan–Cos rule, only tan and cot are positive in QIII, so cosine is negative there: $\cos200^\circ<0$.

Step 2 — locate $320^\circ$. Since $270^\circ<320^\circ<360^\circ$, this angle is in the fourth quadrant, where only cos and sec are positive, so tangent is negative: $\tan320^\circ<0$.

Check (independent route — write each as $180^\circ$ or $360^\circ$ plus/minus a reference angle and use the reduction formula). $200^\circ=180^\circ+20^\circ$, so $\cos200^\circ=\cos(180^\circ+20^\circ)=-\cos20^\circ$, and since $\cos20^\circ$ is a small positive number, $-\cos20^\circ$ is negative — confirming $\cos200^\circ<0$. Similarly $320^\circ=360^\circ-40^\circ$, so $\tan320^\circ=\tan(360^\circ-40^\circ)=-\tan40^\circ$, which is negative — confirming $\tan320^\circ<0$.

Misplacing an angle in the wrong quadrant by miscounting the $90^\circ$ boundaries (e.g. treating $200^\circ$ as QII instead of QIII); also confusing which pair of ratios is positive in QIII versus QIV.