Find the angle at which the line y=0.5 intersects

Find the angle at which the line $y=0.5$ intersects the curve $y=\cos x$.

Step 1: Intersection points

$\cos x=\tfrac12 \Rightarrow x=\pm\dfrac\pi3+2\pi n$; by periodicity, work at $x_0=\dfrac\pi3$.

Step 2: Slopes

The line has slope $0$. The curve has $y'=-\sin x$, so at $x_0=\dfrac\pi3$: $k=-\sin\dfrac\pi3=-\dfrac{\sqrt3}{2}$.

Step 3: Angle

\[ \tan\theta = \left|\frac{-\dfrac{\sqrt3}{2}-0}{1+0}\right| = \frac{\sqrt3}{2}. \tag{1} \]

Step 4: Verification

\[\text{At } x_0=-\tfrac\pi3\text{: } k=\tfrac{\sqrt3}{2},\ \tan\theta=\tfrac{\sqrt3}{2}\ \checkmark \text{ (same angle at every crossing)}\]