Why do we only consider positive angles between two lines?

Question

Whenever we draw a diagram that contains angles, we only take positive angles. For example, we only use the first diagram below and say that $AB$ equals $PA\sin60⁰.$

Why don't we consider negative angles? For example, using the second diagram gives $-PA\sin60⁰.$

Also if consider we might get different answers, right?

Answer 1

You are asking why your example exercise uses undirected angles rather than directed angles.

Technically, your given exercise contains not lines but line segments. My answer to your previous question points out that angles between lines are defined on $[0^\circ,90^\circ],$ whereas angles in a triangle are defined on $(0^\circ,180^\circ).$ When it is unnecessary to consider direction, as in your example, this convention just keeps things simple. If you insist on writing $-60^\circ$ instead of $60^\circ,$ the Sine Rule, for example, will still work as long as you consistently use the negative measure for every angle.

When testing the divisibility of a number $x$ by $6,$ why don't we perform x÷(−2) and x÷(−3) instead of just x÷(2) and x÷(3)? Because the simpler way gives the same result.

When determining the distance travelled, why don't we compute average velocity × time instead of average speed × time? Because using the unsigned quantity speed gives the correct answer while using the signed quantity velocity gives the resultant displacement instead (notice that starting and ending at the same position means that the resultant displacement is zero even after having travelled $100$km).

Answer 2

I think there are 3 things you need to notice:

• the $|AB|$ is a length, which cannot be negative, therefore I think you may need absolute value. As for its geometric meaning, I suppose you can reflect the triangle with respect to the x axis, then hoepfully you'll see the geometric meaning of $-PA\sin60^\circ$;
• conventionally, we use anti-clockwise as the increasing direction of an angle;
• of course, you can use the $-60^\circ$, but it will change your coordinate system. Originally, we are regarding $P$ as the origin, and $AP$ is the hypotenuse of the right-angled triangle. But if you use $-60^\circ$, the hypotenuse of the right-angled $\triangle PBC$ triangle then becomes $BP$, where $BC$ is the altitude with respect to $B$. Subsequently, the meaning of $-PA\sin60^\circ$ has changed.