Tangent Angle to Parametric Ellipse

Question

I have an ellipse defined by the parametric equation: $$(x,y)=(a\cos\theta,b\sin\theta)$$ In this example: $$a=45$$ $$b=15$$ $$\theta =20°$$ How do I calculate the angle of the tangent line?

Answer 1

The slope is $$\frac{dy}{dx} = \frac{dy/d\theta}{dx/d\theta} = -\frac{b\cos\theta}{a\sin\theta}.$$ With the particular values, one gets $$-\frac{15\cos\frac{\pi}{9}}{45\sin\frac{\pi}{9}} \approx -0.915826.$$ This is the slope, so the angle to the positive $x$-axis is $\tan^{-1}(-0.915826) \approx -0.74149 \approx -42.4843^{\circ}$.

Edit: With this Mathematica code

Show[ContourPlot[x^2/45^2 + y^2/15^2 == 1, {x, -45, 45}, {y, -15, 15}, AspectRatio -> 1/3], 
Plot[15 Sin[20 Degree] - 0.915826 (x - 45 Cos[20 Degree]), {x, 0, 55}]]

I get the following plot: