91Ó°ÊÓ

Evaluate \(\sin \frac{7 \pi}{12}\) as \(\sin \left(\frac{\pi}{4}+\frac{\pi}{3}\right)\)

Do not fall into the trap \(|-a|=a .\) For what real numbers \(a\) is this equation true? For what real numbers is it false?

Solve the equation \(|x-1|=1-x\)

Evaluate \(\cos \frac{11 \pi}{12}\) as \(\cos \left(\frac{\pi}{4}+\frac{2 \pi}{3}\right)\)

In Exercises \(41-46,\) find an equation for the circle with the given center \(C(h, k)\) and radius \(a\) . Then sketch the circle in the \(x y\) -plane. Include the circle's center in your sketch. Also, label the circle's \(x\) - and \(y\) -intercepts, if any, with their coordinate pairs. $$ C(1,1), \quad a=\sqrt{2} $$

Evaluate \(\cos \frac{\pi}{12}\)

A proof of the triangle inequality Give the reason justifying each of the numbered steps in the following proof of the triangle inequality. $$ \begin{aligned}|a+b|^{2} &=(a+b)^{2} \\ &=a^{2}+2 a b+b^{2} \\ & \leq a^{2}+2|a||b|+b^{2} \\ &=|a|^{2}+2|a||b|+|b|^{2} \\ &=(|a|+|b|)^{2} \\\|a+b| & \leq|a|+|b| \end{aligned} $$

In Exercises \(41-46,\) find an equation for the circle with the given center \(C(h, k)\) and radius \(a\) . Then sketch the circle in the \(x y\) -plane. Include the circle's center in your sketch. Also, label the circle's \(x\) - and \(y\) -intercepts, if any, with their coordinate pairs. $$ C(-\sqrt{3},-2), \quad a=2 $$

Prove that \(|a b|=|a||b|\) for any numbers \(a\) and \(b\)

Evaluate \(\sin \frac{5 \pi}{12}\)