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i have one question suppose there is given two line by the tangent form $y=3x-2$ and second line $y=5x+3$ we are asked to find smallest positive angle bewteen them in genaral i want to know how to solve such problem for more general situation or when lines are given by $y=ax+b$ and $y=cx+d.$ i am confused what does mean smallest positive angle i know how calculate angle between two line but smallest positive angle confuses me please help.

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hey do you mean y=3(x-2) or $y=3^{x}-2$ – user9413 Jul 8 '11 at 13:15
no y=3x-2 sorry for bad form – dato datuashvili Jul 8 '11 at 13:16
These two lines intersect at some point. since $5x+3=3x-2$ gives a value of $x$. I don't know what smallest angle means, here. – user9413 Jul 8 '11 at 13:19
@Chandru: When two lines intersect, but are not equal, they form four angles. Either all four angles are right angles, or two of the angles are acute (and congruent to one another) and the other two are obtuse (and congruent to one another), so the acute angles have smaller measure than the obtuse angles. – Isaac Jul 8 '11 at 13:30
@Isaac: Ah, right. I Understand what you mean now. – user9413 Jul 8 '11 at 13:58
up vote 3 down vote accepted

Your lines $y=ax+b$ and $y=cx+d$, assuming that they are not horizontal, form angles of directed measure $\arctan(a)$ and $\arctan(c)$ with the $x$-axis (where these angles are between $-90°$ and $90°$, and where horizontal lines will give angle measure $0°$), so an angle between the two lines has measure $\theta=|\arctan(a)-\arctan(c)|$. If $0°\le\theta\le90°$, then this is the smaller angle (or, when $\theta=90°$ the two angles are equal); otherwise, $180°-\theta$ is the smaller angle. If the two lines are parallel or the same line, then $\theta=0°$, which is a sensible answer in that case.

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Two intersecting unequal lines in general determine two sizes of angle at their intersection point.

One size is "small", less than or equal to $90^\circ$. The other angle between the two lines is the "supplementary angle" to the small one. If $\theta$ is the size of the small one, the size of the supplementary angle is $180^\circ -\theta$. (If the lines are perpendicular then the "small" and "big" angle are equal in size.)

Usually, at the informal level, when we talk about the angle at which two lines meet, we choose the small one. It is common to say that the angle between two lines is $30^\circ$, and less common to say that the angle is $150^\circ$.

The author of the problem just wanted specify the angle completely.

The author seems implicitly to allow the possibility of negative angles between two lines, and maybe even large negative angles (travel clockwise from one line to the other, making a few revolutions).

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so as i think tan(q)=(m2-m1)/(1+m1m2) where m1 and m2 are slopes of lines it means that tan(q)=1/8 and q is equal arctan(1/8) – dato datuashvili Jul 8 '11 at 13:29
@user3196: Yes, just fine, since arctan of a positive number will give you an answer in the right range. – André Nicolas Jul 8 '11 at 13:35

Let $\alpha \in ]-\frac{\pi}{2};\frac{\pi}{2}[$ be the angle formed by a non-vertical line and the positive x-axis, then the slope of the first line is equal to $\tan \alpha$ ( why ? ), with that you can easily deduce $\alpha$ because $\alpha = \arctan m$ where $m$ is the slope of your line. You can then use that result to solve your problem.

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