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Chapter 6: Problem 87
What is a directed line segment?
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Without using symbols, state the Law of Cosines in your own words.
Find all the complex roots. Write roots in rectangular form. If necessary, round to the nearest tenth. The complex fourth roots of \(81\left(\cos \frac{4 \pi}{3}+i \sin \frac{4 \pi}{3}\right)\)
Show that the given complex number \(z\) plots as a point in the Mandelbrot set. a. Write the first six terms of the sequence \(z_{1}, z_{2}, z_{3}, z_{4}, z_{5}, z_{6}, \dots\) where \(z_{1}=z:\) Write the given number. \(z_{2}=z^{2}+z:\) Square \(z_{1}\) and add the given number. \(z_{3}=\left(z^{2}+z\right)^{2}+z:\) Square \(z_{2}\) and add the given number. \(z_{4}=\left[\left(z^{2}+z\right)^{2}+z\right]^{2}+z:\) Square \(z_{3}\) and add the given number. \(z_{5}:\) Square \(z_{4}\) and add the given number. \(z_{6}:\) Square \(z_{5}\) and add the given number. b. If the sequence that you began writing in part (a) is bounded, the given complex number belongs to the Mandelbrot set. Show that the sequence is bounded by writing two complex numbers. One complex number should be greater in absolute value than the absolute values of the terms in the sequence. The second complex number should be less in absolute value than the absolute values of the terms in the sequence. $$z=-i$$
A pine tree growing on a hillside makes a \(75^{\circ}\) angle with the hill. From a point 80 feet up the hill, the angle of elevation to the top of the tree is \(62^{\circ}\) and the angle of depression to the bottom is \(23^{\circ} .\) Find, to the nearest tenth of a foot, the height of the tree. (Figure can't copy)
I'm working with a unit vector, so its dot product with itself must be 1
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