Practice (117)

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Without using a calculator, find the value of $\cos\frac{\pi}{13}+\cos\frac{3\pi}{13}+\cos\frac{9\pi}{13}$.

Given two segments $AB$ and $MN$, show that $$MN\perp AB \Leftrightarrow AM^2 - BM^2 = AN^2 - BN^2$$

In $\triangle{ABC}$, let $a$, $b$, and $c$ be the lengths of sides opposite to $\angle{A}$, $\angle{B}$ and $\angle{C}$, respectively. $D$ is a point on side $AB$ satisfying $BC=DC$. If $AD=d$, show that $$c+d=2\cdot b\cdot\cos{A}\quad\text{and}\quad c\cdot d = b^2-a^2$$

Let $f(x) = \sin{x} + 2\cos{x} + 3\tan{x}$, using radian measure for the variable $x$. In what interval does the smallest positive value of $x$ for which $f(x) = 0$ lie?

Prove the triple angle formulas: $$\sin 3\theta = 3\sin\theta -4\sin^3\theta$$ and $$\cos 3\theta = 4\cos^3\theta - 3\cos\theta$$

Show that $\sin\alpha + \sin\beta + \sin\gamma - \sin(\alpha + \beta+\gamma) = 4\sin\frac{\alpha+\beta}{2}\sin\frac{\beta+\gamma}{2}\sin\frac{\gamma+\alpha}{2}$

Compute $\cot 70^\circ + 4\cos 70^\circ$

$\displaystyle\frac{2\cos 2^n A+1}{2\cos A+1}=\prod_{r=1}^{n} (2\cos 2^{r-1}A-1)$

Compute $4\cos\frac{2\pi}{7}\cos\frac{\pi}{7}-2\cos\frac{2\pi}{7}$


Find an acute angle $\alpha$ such that $\sqrt{15-12\cos\alpha} + \sqrt{7-4\sqrt{3}\sin\alpha}=4$. (Find at least two different solutions.)

Explain that $$\sin x +\sin (x+120^\circ) + \sin (x-120^\circ) = \cos x +\cos (x+120^\circ) + \cos (x-120^\circ) = 0$$ using at least two approaches.

Compute the value of $\sin 1^\circ \sin 2^\circ \cdots \sin 89^\circ$.

If $0 < \alpha < \beta < \frac{\pi}{2}$, show $$\frac{\cot\beta}{\cot\alpha}<\frac{\cos\beta}{\cos\alpha}<\frac{\beta}{\alpha}$$

Evaluate $\cos\frac{\pi}{2n+1}+\cos\frac{3\pi}{2n+1}+\cdots+\cos\frac{(2n-1)\pi}{2n+1}$.


In $\triangle{ABC}$ show that $$\tan\frac{A}{2}\tan\frac{B}{2}+\tan\frac{B}{2}\tan\frac{C}{2}+\tan\frac{C}{2}\tan\frac{A}{2}=1$$

In $\triangle{ABC}$, show that $$\tan\frac{A}{2}\tan\frac{B}{2}\tan\frac{C}{2}\le\frac{\sqrt{3}}{9}$$

In $\triangle{ABC}$, $\angle{C}=\angle{A}+60^\circ$. If $BC=1$, $AC=r$ and $AB=r^2$, where $r > 1$, prove $r \le\sqrt{2}$.

In $\triangle{ABC}$, show that \begin{equation} \sin A + \sin B + \sin C = 4\cos\frac{A}{2}\cos\frac{B}{2}\cos\frac{C}{2} \end{equation} Try to use at least two different approaches.

In $\triangle{ABC}$ show that $\cos A +\cos B + \cos C \le\frac{3}{2}$.

Given any $\triangle{ABC}$, show that $$\cos A + \cos B + \cos C = 1+4\sin\frac{A}{2}\sin\frac{B}{2}\sin\frac{C}{2}$$

In $\triangle{ABC}$, show that $$\sin\frac{A}{2}=\sqrt{\frac{(p-b)(p-c)}{bc}}$$ where $p=\frac{a+b+c}{2}$ is the semi-perimeter.

Show that $$S_{\triangle{ABC}}=\frac{abc}{4R}$$

Show that $$\sec^2\alpha = 1 + \tan^2\alpha$$ $$\csc^2\alpha = 1 + \cot^2\alpha$$

Prove the identity: $\tan^2 x - \sin^2 x = \tan^2 x \sin^2 x$.

If $\cos x - \sin x = \sqrt{2}\sin x$, prove $\cos x +\sin x = \sqrt{2}\cos x$.