grandes-ecoles 2019 Q15

grandes-ecoles · France · x-ens-maths__pc Trig Proofs Trigonometric Inequality Proof

Let $\theta \in [-\pi, \pi]$.
a. Show that $\cos(\theta) \geq 1 - \frac{\theta^2}{2}$.
b. Show that $\left|\frac{e^{i\theta} - (1-p)}{p}\right| \leq \exp\left(\frac{1-p}{2p^2} \cdot \theta^2\right)$.
Hint. One may calculate $\left|\frac{e^{i\theta} - (1-p)}{p}\right|^2$.

Let $\theta \in [-\pi, \pi]$.

a. Show that $\cos(\theta) \geq 1 - \frac{\theta^2}{2}$.

b. Show that $\left|\frac{e^{i\theta} - (1-p)}{p}\right| \leq \exp\left(\frac{1-p}{2p^2} \cdot \theta^2\right)$.

Hint. One may calculate $\left|\frac{e^{i\theta} - (1-p)}{p}\right|^2$.

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