Tangent lines to parametric curves
Problem 8.15 · hard
Find the equation of the tangent line to \( \displaystyle x = \cos{\left(t \right)},\ y = \sin{\left(t \right)} \) at \( \displaystyle t = \frac{\pi}{3} \).
- dy/dx = (dy/dt)/(dx/dt).
- \[ \frac{d}{d t} \cos{\left(t \right)} = - \sin{\left(t \right)} \]dx/dt.✓ Proved
- \[ \frac{d}{d t} \sin{\left(t \right)} = \cos{\left(t \right)} \]dy/dt.✓ Proved
- \[ \frac{\left(-1\right) \sqrt{3}}{3} = - \frac{\sqrt{3}}{3} \]The slope at t = \frac{\pi}{3}.✓ Proved
- \[ \left[\begin{matrix}1 \cdot \frac{1}{2}\\\frac{\sqrt{3}}{2}\end{matrix}\right] = \left[\begin{matrix}\frac{1}{2}\\\frac{\sqrt{3}}{2}\end{matrix}\right] \]The point.✓ Proved
- \[ - \frac{\sqrt{3} \left(x - \frac{1}{2}\right)}{3} + \frac{\sqrt{3}}{2} = - \frac{\sqrt{3} x}{3} + \frac{2 \sqrt{3}}{3} \]Point-slope form.✓ Proved
Answer \( y = - \frac{\sqrt{3} x}{3} + \frac{2 \sqrt{3}}{3} \)
Lines: 5 proved, 1 not checked. The answer was also checked a second way, without looking at the solution. The explanation has not been reviewed yet.
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | Not checked | — | a sentence; read, not computed |
| 2 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | a secant through nearby points of the curve has the same slope; the line passes through the point |
Proved: SymPy reduced the difference between the two sides to zero. Checked independently: a separate
method, named above, confirmed it. Checked numerically: the two sides agree at every sampled point, which is evidence,
not proof. Reviewed: a model or a person read it; that is all a sentence can have. Solution by
generator:structured/parametric_tangent, checked 2026-09-26 with SymPy 1.14.0.