91Ó°ÊÓ

Explain why the Pythagorean Theorem is a special case of the Law of Cosines.

Use DeMoivre's Theorem to find the indicated power of the complex number. Write answers in rectangular form. $$\left[\sqrt{3}\left(\cos \frac{5 \pi}{18}+i \sin \frac{5 \pi}{18}\right)\right]^{6}$$

In Exercises \(57-60,\) let $$\begin{aligned}\mathbf{u} &=a_{1} \mathbf{i}+b_{1} \mathbf{j} \\\\\mathbf{v} &=a_{2}\mathbf{i}+b_{2} \mathbf{j} \\\\\mathbf{w} &=a_{3} \mathbf{i}+b_{3} \mathbf{j}\end{aligned}$$ Prove each property by obtaining the vector on each side of the equation. Have you proved a distributive, associative, or commutative property of vectors? $$(c+d) \mathbf{u}=c \mathbf{u}+d \mathbf{u}$$

Convert each polar equation to a rectangular equation. Then use a rectangular coordinate system to graph the rectangular equation. $$\theta=\frac{\pi}{2}$$

What is an oblique triangle?

Use a graphing utility to graph the polar equation. $$r=4 \sin 6 \theta$$

Use DeMoivre's Theorem to find the indicated power of the complex number. Write answers in rectangular form. $$(1+i)^{5}$$

A force of 4 pounds acts in the direction of \(50^{\circ}\) to the horizontal. The force moves an object along a straight line from the point (3,7) to the point \((8,10),\) with distance measured in feet. Find the work done by the force.

In Exercises \(61-64,\) find the magnitude \(\|\mathbf{v}\|,\) to the nearest hundredth, and the direction angle \(\theta,\) to the nearest tenth of \(a\) degree, for each given vector \(\mathbf{v}\). $$\mathbf{v}=-10 \mathbf{i}+15 \mathbf{j}$$

Without using symbols, state the Law of sines in your own words.