Equations

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Finding solutions to trigonometric equations.

As a general rule, since this is a review assignment, you should probably do this several times before you actually take the exam. The actual goal of this is to make sure that you can pinpoint the spots where you are not as prepared as you should be going in. Doing this exactly once will probably not be as helpful as you would like it to be unless you sit down, do all of them perfectly on the first try AND the actual longest part of the experience is putting in your answer.

The formatting on these answers is particular. Any given answer is going to be a list, or even multiple lists, of infinitely many angles, so answers have been formatted in a consistent way. Within a row, the first entry should be the least nonnegative angle in the list. The second entry should just be whatever is being multiplied by $n$ to get the rest of the answers in the list. The rows themselves are ordered so that the first entries of each row increase from top to bottom.

For example, if we had $x = \frac {\pi }{3} + 2\pi \cdot n, -\frac {\pi }{3} + 2\pi \cdot n$ , then we would write this as $x = \begin {cases} \tfrac {\pi }{3} + 2\pi \cdot n \\ \tfrac {5\pi }{3} + 2\pi \cdot n \end {cases}$

Notice that both of the first entries are nonnegative (we had to use $\tfrac {5\pi }{3}$ instead of $-\tfrac {\pi }{3}$ ) and are in ascending order.

1 : For this first problem, we focus on the most basic expressions. Thing of the form trig $(x)=y$ and try to find all values of $x$ satisfying the equation. You will want to make sure that you are very solid on this version of the problem before moving on as completing later sections of this assignment will hinge on reducing each problem to things of this form.

Note: Even though the period of tangent and cotanget is $pi$ this problem will still want all families of solutions in terms of $[0,2pi)$ , we’ll worry about condensing those down to one family a little later.

(a): $\sage {Fun1a}=\sage {Root1a}$ $x = \begin {cases} \answer {\sage {Ang1aS}} + \answer {\sage {Per1a}} \cdot n \\ \answer {\sage {Ang1aB}} + \answer {\sage {Per1a}} \cdot n \end {cases}$
(b): $\sage {Fun1b}=\sage {Root1b}$ $x = \begin {cases} \answer {\sage {Ang1bS}} + \answer {\sage {Per1b}} \cdot n \\ \answer {\sage {Ang1bB}} + \answer {\sage {Per1b}} \cdot n \end {cases}$
(c): $\sage {Fun1c}=\sage {Root1c}$ $x = \begin {cases} \answer {\sage {Ang1cS}} + \answer {\sage {Per1c}} \cdot n \\ \answer {\sage {Ang1cB}} + \answer {\sage {Per1c}} \cdot n \end {cases}$
(d): $\sage {Fun1d}=\sage {Root1d}$ $x = \begin {cases} \answer {\sage {Ang1dS}} + \answer {\sage {Per1d}} \cdot n \\ \answer {\sage {Ang1dB}} + \answer {\sage {Per1d}} \cdot n \end {cases}$
(e): $\sage {Fun1e}=\sage {Root1e}$ $x = \begin {cases} \answer {\sage {Ang1eS}} + \answer {\sage {Per1e}} \cdot n \\ \answer {\sage {Ang1eB}} + \answer {\sage {Per1e}} \cdot n \end {cases}$

2 : These problems are secretly the same as the previous set. The difference is: instead of starting from $x=\tfrac {a-b}{c}$ you are starting from a point that looks more like $cx+b=a$ . The goal is to become comfortable reducing expressions of this type to expressions of the previous type.

(a): $\sage {LH2a}=\sage {RH2a}$ $x = \begin {cases} \answer {\sage {Ang2aS}} + \answer {\sage {Per2a}} \cdot n \\ \answer {\sage {Ang2aB}} + \answer {\sage {Per2a}} \cdot n \end {cases}$
(b): $\sage {LH2b}=\sage {RH2b}$ $x = \begin {cases} \answer {\sage {Ang2bS}} + \answer {\sage {Per2b}} \cdot n \\ \answer {\sage {Ang2bB}} + \answer {\sage {Per2b}} \cdot n \end {cases}$

3 : These problems build on the previous sets, but with a little more collecting terms necessary to complete the problem.

(a): $\sage {LH3a}=\sage {RH3a}$ $x = \begin {cases} \answer {\sage {Ang3aS}} + \answer {\sage {Per3a}} \cdot n \\ \answer {\sage {Ang3aB}} + \answer {\sage {Per3a}} \cdot n \end {cases}$
(b): $\sage {LH3b}=\sage {RH3b}$ $x = \begin {cases} \answer {\sage {Ang3bS}} + \answer {\sage {Per3b}} \cdot n \\ \answer {\sage {Ang3bB}} + \answer {\sage {Per3b}} \cdot n \end {cases}$

4 : Remembering that if you have something of the form $(x-a)(x-b)=0$ the solutions $x=a$ or $x=b$ , find all possible values for $x$ satisfying the following equations.

(a): $\sage {Disp4a} = 0$ $x = \begin {cases} \answer {\sage {Ans4a11}} + \answer {\sage {Ans4a12}} \cdot n \\ \answer {\sage {Ans4a21}} + \answer {\sage {Ans4a22}} \cdot n \\ \answer {\sage {Ans4a31}} + \answer {\sage {Ans4a32}} \cdot n \\ \answer {\sage {Ans4a41}} + \answer {\sage {Ans4a42}} \cdot n \end {cases}$
(b): $\sage {Disp4b} = 0$ $x = \begin {cases} \answer {\sage {Ans4b11}} + \answer {\sage {Ans4b12}} \cdot n \\ \answer {\sage {Ans4b21}} + \answer {\sage {Ans4b22}} \cdot n \\ \answer {\sage {Ans4b31}} + \answer {\sage {Ans4b32}} \cdot n \\ \answer {\sage {Ans4b41}} + \answer {\sage {Ans4b42}} \cdot n \end {cases}$
(c): $\sage {Disp4c} = 0$ $x = \begin {cases} \answer {\sage {Ans4c11}} + \answer {\sage {Ans4c12}} \cdot n \\ \answer {\sage {Ans4c21}} + \answer {\sage {Ans4c22}} \cdot n \\ \answer {\sage {Ans4c31}} + \answer {\sage {Ans4c32}} \cdot n \\ \answer {\sage {Ans4c41}} + \answer {\sage {Ans4c42}} \cdot n \end {cases}$
(d): $\sage {Disp4d} = 0$ $x = \begin {cases} \answer {\sage {Ans4d11}} + \answer {\sage {Ans4d12}} \cdot n \\ \answer {\sage {Ans4d21}} + \answer {\sage {Ans4d22}} \cdot n \\ \answer {\sage {Ans4d31}} + \answer {\sage {Ans4d32}} \cdot n \\ \answer {\sage {Ans4d41}} + \answer {\sage {Ans4d42}} \cdot n \end {cases}$

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shift+ctrl+left/right	Add copy of current entry/column to to list/matrix
ctrl+up/down	Add row to matrix
shift+ctrl+up/down	Add copy of current row to matrix
ctrl+backspace	Delete current entry in list or column in matrix
ctrl+shift+backspace	Delete current row in matrix

Type...	...to get
norm	$\|\|\blue{[?]}\|\|$
text	$\text{\blue{[?]}}$
sym_name	$\backslash\texttt{\blue{[?]}}$
abs	$\left\|\blue{[?]}\right\|$
sqrt	$\sqrt{\blue{[?]}}$
paren	$\left(\blue{[?]}\right)$
floor	$\lfloor \blue{[?]} \rfloor$
factorial	$\blue{[?]}!$
exp	${\blue{[?]}}^{\blue{[?]}}$
sub	${\blue{[?]}}_{\blue{[?]}}$
frac	$\dfrac{\blue{[?]}}{\blue{[?]}}$
int	$\displaystyle\int{\blue{[?]}}d\blue{[?]}$
defi	$\displaystyle\int_{\blue{[?]}}^{\blue{[?]}}\blue{[?]}d\blue{[?]}$
deriv	$\displaystyle\frac{d}{d\blue{[?]}}\blue{[?]}$
sum	$\displaystyle\sum_{\blue{[?]}}^{\blue{[?]}}\blue{[?]}$
prod	$\displaystyle\prod_{\blue{[?]}}^{\blue{[?]}}\blue{[?]}$
root	$\sqrt[\blue{[?]}]{\blue{[?]}}$
vec	$\left\langle \blue{[?]} \right\rangle$
mat	$\left(\begin{matrix} \blue{[?]} \end{matrix}\right)$
*	$\cdot$
infinity	$\infty$
arcsin	$\arcsin\left(\blue{[?]}\right)$
arccos	$\arccos\left(\blue{[?]}\right)$
arctan	$\arctan\left(\blue{[?]}\right)$
sin	$\sin\left(\blue{[?]}\right)$
cos	$\cos\left(\blue{[?]}\right)$
tan	$\tan\left(\blue{[?]}\right)$
sec	$\sec\left(\blue{[?]}\right)$
csc	$\csc\left(\blue{[?]}\right)$
cot	$\cot\left(\blue{[?]}\right)$
log	$\log\left(\blue{[?]}\right)$
ln	$\ln\left(\blue{[?]}\right)$
alpha	$\alpha$
beta	$\beta$
gamma	$\gamma$
delta	$\delta$
epsilon	$\epsilon$
zeta	$\zeta$
eta	$\eta$
theta	$\theta$
iota	$\iota$
kappa	$\kappa$
lambda	$\lambda$
mu	$\mu$
nu	$\nu$
xi	$\xi$
omicron	$\omicron$
pi	$\pi$
rho	$\rho$
sigma	$\sigma$
tau	$\tau$
upsilon	$\upsilon$
phi	$\phi$
chi	$\chi$
psi	$\psi$
omega	$\omega$
Gamma	$\Gamma$
Delta	$\Delta$
Theta	$\Theta$
Lambda	$\Lambda$
Xi	$\Xi$
Pi	$\Pi$
Sigma	$\Sigma$
Phi	$\Phi$
Psi	$\Psi$
Omega	$\Omega$

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