How to Find the Acute Angle Between Two Curves at Their First Intersection

The angle between two curves at an intersection is defined as the angle between their tangent lines at that point. To find it, first identify the intersection in the specified domain or direction, then calculate the slope of each curve there.

The phrase “first intersection” needs a domain or a direction of travel to be unambiguous. For example, it could mean the intersection with the smallest $x$-coordinate, or the first one reached as $x$ increases from a stated starting value.

The tangent-line angle formula

Suppose the curves are graphs of differentiable functions $y=f(x)$ and $y=g(x)$, and they intersect at $x=a$. Their tangent slopes are

$$m_1=f'(a), \qquad m_2=g'(a).$$

For two nonvertical lines with slopes $m_1$ and $m_2$, the acute angle $\theta$ between them satisfies

$$\tan\theta=\left|\frac{m_2-m_1}{1+m_1m_2}\right|.$$

Thus, when $1+m_1m_2\ne 0$,

$$\theta=\arctan\left(\left|\frac{m_2-m_1}{1+m_1m_2}\right|\right),$$

where $\arctan$ returns an angle between $0$ and $\pi/2$ for a nonnegative input. To convert to degrees, multiply the result in radians by $180/\pi$.

If $1+m_1m_2=0$, the tangent lines are perpendicular, so their angle is $90^\circ$ (or $\pi/2$ radians). Although this is often included when people say “the acute angle,” it is technically a right angle, not an acute one.

Step-by-step method

1. Set the function values equal. Solve $f(x)=g(x)$ to find all intersection coordinates in the relevant domain.
2. Choose the first intersection. Use the direction or starting point given in the problem. If none is specified, say which convention you are using, such as the smallest $x$-coordinate.
3. Differentiate both functions. Find $f'(x)$ and $g'(x)$.
4. Evaluate the derivatives at the intersection. These values are the slopes of the two tangent lines.
5. Calculate the angle. Apply the slope formula, checking separately for vertical tangents or perpendicular tangent lines.

Worked example

Find the angle between $y=x^2$ and $y=1$ at their first intersection for $x\ge 0$, moving from left to right.

First, solve for the intersections:

$$x^2=1 \quad\Longrightarrow\quad x=-1 \text{ or } x=1.$$

On the domain $x\ge 0$, the first intersection is $x=1$. The point is $(1,1)$.

Differentiate the functions:

$$\frac{d}{dx}(x^2)=2x, \qquad \frac{d}{dx}(1)=0.$$

At $x=1$, the tangent slopes are $m_1=2$ and $m_2=0$. Therefore,

$$\tan\theta=\left|\frac{0-2}{1+(2)(0)}\right|=2, \qquad \theta=\arctan(2)\approx 63.4^\circ.$$

So the tangent lines—and hence the curves at that point—meet at an angle of about $63.4^\circ$.

Vertical tangents and a general method

The slope formula assumes both tangent lines have finite slopes. If one tangent is vertical, or if the curves are given parametrically, use tangent direction vectors instead.

For tangent vectors $\mathbf{u}$ and $\mathbf{v}$, the smaller angle between the tangent lines is

$$\theta=\arccos\left(\frac{|\mathbf{u}\cdot\mathbf{v}|}{\|\mathbf{u}\|\,\|\mathbf{v}\|}\right).$$

The absolute value selects the smaller angle between the lines, which lies between $0$ and $90^\circ$. For a graph $y=f(x)$ with a finite derivative at $x=a$, a tangent direction vector is $(1,f'(a))$. A vertical tangent can instead be represented by a direction vector such as $(0,1)$.

For instance, a line with slope $2$ has direction vector $(1,2)$, and a vertical line has direction vector $(0,1)$. Their smaller angle is

$$\arccos\left(\frac{|(1,2)\cdot(0,1)|}{\sqrt{5}\cdot 1}\right) =\arccos\left(\frac{2}{\sqrt{5}}\right) \approx 26.6^\circ.$$

Common mistakes

• Finding an intersection but not the first one. Solve for every intersection, then apply the domain and direction condition. “First” is not meaningful until those conditions are specified.
• Using function values instead of derivatives. The angle is determined by the tangent slopes at the intersection, not by the heights or the values of the functions.
• Forgetting the absolute value. Without it, the calculation can give a signed angle or an obtuse angle. The acute angle between tangent lines uses the smaller, nonnegative angle.
• Applying the slope formula to a vertical tangent. A vertical tangent has no finite slope. Use tangent direction vectors, or take a limit when the tangent direction is well-defined.
• Assuming a corner has one tangent. If a curve is not differentiable at the intersection, it may have no unique tangent direction. Check its one-sided tangent directions before assigning an angle.

If the two tangent slopes are equal, the angle is $0^\circ$: the curves share a tangent there, even if they cross at that point.