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Mathematics LibreTexts

6.E: Additional Topics (Exercises)

6.1 Exercise


For Exercises 1-12, solve the given equation (in radians).

6.1.1  \(\tan\;\theta \;+\; 1 ~=~ 0\)

6.1.2  \(2\,\cos\;\theta \;+\; 1 ~=~ 0\)

6.1.3  \(\sin\;5\theta \;+\; 1 ~=~ 0\)

6.1.4  \(2\,\cos^2 \;\theta \;-\; \sin^2 \;\theta ~=~ 1\)

6.1.5  \(2\,\sin^2 \;\theta \;-\; \cos\;2\theta ~=~ 0\)

6.1.6  \(2\,\cos^2 \;\theta \;+\; 3\,\sin\;\theta ~=~ 0\)

6.1.7  \(\cos^2 \;\theta \;+\; 2\,\sin\;\theta ~=~ -1\)

6.1.8  \(\tan\;\theta \;+\; \cot\;\theta ~=~ 2\)

6.1.9  \(\sin\;\theta ~=~ \cos\;\theta\)

6.1.10  \(2\,\sin\;\theta \;-\; 3\,\cos\;\theta ~=~ 0\)

6.1.11  \(\cos^2 \;3\theta \;-\; 5\,\cos\;3\theta \;+\; 4 ~=~ 0\)

6.1.12  \(3\,\sin\;\theta \;-\; 4\,\cos\;\theta ~=~ 1\)
 


6.2 Exercise


6.2.1  One obvious solution to the equation \(2\,\sin\;x = x \) is \(x=0 \). Write a program to find the other solution(s), accurate to at least within \(1.0 \,\times\, 10^{-20} \). You can use any programming language, though you may find it easier to just modify the code in Listing 6.1 (only one line needs to be changed!). It may help to use Gnuplot to get an idea of where the graphs of \(y=2\,\sin\;x \) and \(y=x\) intersect.

6.2.2  Repeat Exercise 1 for the equation \(\sin\;x = x^2 \).

6.2.3  Use Octave or some other program to find the maximum and minimum of \(y=\cos\;5x - \sin\;3x \).
 


6.3 Exercise

For Exercises 1-16, calculate the given expression.

6.3.1  \((2+3i) \;+\; (-3-2i)\)

6.3.2  \((2+3i) \;-\; (-3-2i)\)

6.3.3  \((2+3i) \;\cdot\; (-3-2i)\)

6.3.4  \((2+3i)/(-3-2i)\)

6.3.5  \(\overline{(2+3i)} \;+\; \overline{(-3-2i)}\)

6.3.6  \(\overline{(2+3i)} \;-\; \overline{(-3-2i)}\)

6.3.7  \((1+i)/(1-i)\)

6.3.8  \(|-3+2i|\)

6.3.9  \(i^3\)

6.3.10  \(i^4\)

6.3.11  \(i^5\)

6.3.12  \(i^6\)

6.3.13  \(i^7\)

6.3.14  \(i^8\)

6.3.15  \(i^9\)

6.3.16  \(i^{2009}\)

For Exercises 17-24, prove the given identity for all complex numbers.

6.3.17  \(\overline{\left( \overline{z} \right)} \;=\; z\)

6.3.18  \(\overline{z_1 + z_2} \;=\; \overline{z_1} + \overline{z_2}\)

6.3.19  \(\overline{z_1 - z_2} \;=\; \overline{z_1} - \overline{z_2}\)

6.3.20  \(\overline{z_1 \, z_2} \;=\; \overline{z_1} ~ \overline{z_2}\)

6.3.21  \(\overline{\left( \dfrac{z_1}{z_2} \right)} \;=\; \dfrac{\overline{z_1}}{\overline{z_2}}\)

6.3.22  \(|z| \;=\; |\overline{z}|\phantom{\dfrac{|1_1|}{|1_2|}}\)

6.3.23  \(|z_1 \, z_2| \;=\; |z_1|\,|z_2|\phantom{\dfrac{|1_1|}{|1_2|}}\)

6.3.24  \(\left| \dfrac{z_1}{z_2} \right| \;=\; \dfrac{|z_1|}{|z_2|}\)

For Exercises 25-30, put the given number in trigonometric form.

6.3.25  \(2+3i\)

6.3.26  \(-3-2i\)

6.3.27  \(1-i\)

6.3.28  \(-i\)

6.3.29  \(1\)

6.3.30  \(-1\)

6.3.31  Verify that De Moivre's Theorem holds for the power \(n=0 \).

For Exercises 32-35, calculate the given number.

6.3.32   \(3\,(\cos\;14^\circ \;+\; i\,\sin\;14^\circ ) \;\cdot\; 2\,(\cos\;121^\circ \;+\; i\,\sin\;121^\circ )\)

6.3.33  \(\lbrack 3\,(\cos\;14^\circ \;+\; i\,\sin\;14^\circ )\rbrack^4\phantom{\dfrac{3}{4}}\)

6.3.34  \(\lbrack 3\,(\cos\;14^\circ \;+\; i\,\sin\;14^\circ )\rbrack^{-4}\phantom{\dfrac{3}{4}}\)

6.3.35  \(\dfrac{3\,(\cos\;14^\circ \;+\; i\,\sin\;14^\circ )}{2\,(\cos\;121^\circ \;+\; i\,\sin\;121^\circ )}\)

6.3.36  Find the three cube roots of \(-i \).

6.3.37  Find the three cube roots of \(1+i \).

6.3.38  Find the three cube roots of \(1 \).

6.3.39  Find the three cube roots of \(-1 \).

6.3.40  Find the five fifth roots of \(1 \).

6.3.41  Find the five fifth roots of \(-1 \).

6.3.42  Find the two square roots of \(-2 + 2\sqrt{3}\,i \).

6.3.43  Prove that if \(z \) is an \(n^{th} \) root of a real number \(a \), then so is \(\overline{z} \).(Hint: Use Exercise 20.)
 


6.4 Exercise


For Exercises 1-5, convert the given point from polar coordinates to Cartesian coordinates.

6.4.1  \((6,210^\circ)\)

6.4.2  \((-4,3\pi)\)

6.4.3  \((2,11\pi/6)\)

6.4.4  \((6,90^\circ)\)

6.4.5  \((-1,405^\circ)\)

For Exercises 6-10, convert the given point from Cartesian coordinates to polar coordinates.

6.4.6  \((3,1)\)

6.4.7  \((-1,-3)\)

6.4.8  \((0,2)\)

6.4.9  \((4,-2)\)

6.4.10  \((-2,0)\)

For Exercises 11-18, write the given equation in polar coordinates.

6.4.11  \((x-3)^2 + y^2 = 9\)

6.4.12  \(y = -x\)

6.4.13  \(x^2 - y^2 = 1\)

6.4.14  \(3x^2 + 4y^2 - 6x = 9\)

6.4.15  Graph the function \(r = 1 + 2\,\cos\;\theta \) in polar coordinates.