91Ó°ÊÓ

Use set-builder notation to describe the polar region. Assume that the region contains its bounding curves. The region inside the circle \(r=5\) which lies in Quadrant III.

The SS Bigfoot leaves Yeti Bay on a course of \(\mathrm{N} 37^{\circ} \mathrm{W}\) at a speed of 50 miles per hour. After traveling half an hour, the captain determines he is 30 miles from the bay and his bearing back to the bay is \(\mathrm{S} 40^{\circ} \mathrm{E}\). What is the speed and bearing of the ocean current? Round the speed to the nearest mile per hour and express the heading as a bearing, rounded to the nearest tenth of a degree.

60 through #63. In Exercises \(60-63,\) you need to solve for \(\theta\) … # Convert the equation from rectangular coordinates into polar coordinates. Solve for \(r\) in all but #60 through #63. In Exercises \(60-63,\) you need to solve for \(\theta\) \(y=4 x-19\)

In Exercises \(65-76\), find the indicated complex roots. Express your answers in polar form and then convert them into rectangular form. $$ \text { the three cube roots of } z=64 $$

60 through #63. In Exercises \(60-63,\) you need to solve for \(\theta\) … # Convert the equation from rectangular coordinates into polar coordinates. Solve for \(r\) in all but #60 through #63. In Exercises \(60-63,\) you need to solve for \(\theta\) \(x^{2}+y^{2}-2 y=0\)

Convert the equation from polar coordinates into rectangular coordinates. \(r=-3\)

Convert the equation from polar coordinates into rectangular coordinates. \(r=\sqrt{2}\)

According to Theorem 3.16 in Section 3.4 , the polynomial \(p(x)=x^{4}+4\) can be factored into the product linear and irreducible quadratic factors. In Exercise 28 in Section \(8.7,\) we showed you how to factor this polynomial into the product of two irreducible quadratic factors using a system of non-linear equations. Now that we can compute the complex fourth roots of -4 directly, we can simply apply the Complex Factorization Theorem, Theorem \(3.14,\) to obtain the linear factorization \(p(x)=(x-(1+i))(x-(1-i))(x-(-1+i))(x-(-1-i))\). By multiplying the first two factors together and then the second two factors together, thus pairing up the complex conjugate pairs of zeros Theorem 3.15 told us we'd get, we have that \(p(x)=\left(x^{2}-2 x+2\right)\left(x^{2}+2 x+2\right)\). Use the 12 complex \(12^{\text {th }}\) roots of 4096 to factor \(p(x)=x^{12}-4096\) into a product of linear and irreducible quadratic factors.

Convert the equation from polar coordinates into rectangular coordinates. \(r=3 \sin (\theta)\)

Convert the equation from polar coordinates into rectangular coordinates. \(r=7 \sec (\theta)\)