Real Trigonometric Equations#
Definition: Trigonometric Equation
A trigonometric equation is an equation which contains variables in the arguments of real trigonometric functions.
Elementary Trigonometric Equations#
Algorithm: Solving Equations of the Form \(\sin x = c\)
\[ \sin x = c \]
Solutions:
- If \(|c| \gt 1\), then \(x \in \varnothing\).
- If \(c \in [-1;1]\), then
\[ \begin{aligned}x &= \arcsin c + 2k \pi \\ x &= -\arcsin c + (2k+1)\pi \end{aligned} \qquad k \in \mathbb{Z} \]
Algorithm: Solving Equations of the Form \(\cos x = c\)
\[ \cos x = c \]
Solutions:
- If \(|c| \gt 1\), then \(x \in \varnothing\).
- If \(c \in [-1;1]\), then
\[ x =\pm \arccos c + 2k\pi \qquad k \in \mathbb{Z} \]
Algorithm: Solving Equations of the Form \(\tan x = c\)
\[ \tan x = c \]
Requirements:
- \(x \ne \frac{\pi}{2} + k\pi \qquad k \in \mathbb{Z}\)
Solution:
\[ x = \arctan c + k\pi \qquad k \in \mathbb{Z} \]
Algorithm: Solving Equations of the Form \(\cot x = c\)
\[ \cot x = c \]
Requirements:
- \(x \ne k\pi \qquad k \in \mathbb{Z}\)
Solution:
\[ x = \mathop{\operatorname{arccot}} + k\pi \qquad k \in \mathbb{Z} \]
Composed Trigonometric Equations#
Algorithm: Solving Equations of the Form \(\sin f(x) = \sin g(x)\)
We are given a trigonometric equation of the form
\[ \sin f(x) = \sin g(x) \]
Solution:
\[ \begin{aligned}f(x) &= g(x) + 2k\pi \\ f(x) &= -g(x) + (2k+1)\pi \end{aligned} \qquad k \in \mathbb{Z} \]
Algorithm: Solving Equations of the Form \(\cos f(x) = \cos g(x)\)
We are given a trigonometric equation of the form
\[ \cos f(x) = \cos g(x) \]
Solution:
\[ f(x) = \pm g(x) + 2k\pi \qquad k \in \mathbb{Z} \]
Algorithm: Solving Equations of the Form \(\tan f(x) = \tan g(x)\)
We are given a trigonometric equation of the form
\[ \tan f(x) = \tan g(x) \]
Requirements:
- \(f(x),g(x) \ne \frac{\pi}{2} + k\pi \qquad k \in \mathbb{Z}\)
Solution:
\[ f(x) = g(x) + k\pi \qquad k \in \mathbb{Z} \]
Algorithm: Solving Equations of the Form \(\cot f(x) = \cot g(x)\)
We are given a trigonometric equation of the form
\[ \cot f(x) = \cot g(x) \]
Requirements:
- \(f(x), g(x) \ne k\pi \qquad k \in \mathbb{Z}\)
Solution:
\[ f(x) = g(x) + k\pi \qquad k \in \mathbb{Z} \]
Homogeneous Trigonometric Equations#
Definition: Homogeneous Trigonometric Equation
A homogeneous trigonometric equation is a trigonometric equation of the form
\[ a_0 \sin^n x + a_1 \sin^{n-1} x \cos x + \cdots + a_{n-1}\sin x \cos^{n-1} x + a_n \cos^n x = 0, \]
where \(a_k \in \mathbb{R}\).
Algorithm: Solving Homogeneous Trigonometric Equations (Tangent Substitution)
We are given the following [[Homogeneous Trigonometric Equations]].
\[ a_0 \sin^n x + a_1 \sin^{n-1} x \cos x + \cdots + a_{n-1}\sin x \cos^{n-1} x + a_n \cos^n x = 0 \]
- Check whether \(\cos^n (x) = 0\), i.e. \(x = \pm\frac{\pi}{2} + 2k\pi\) for \(k \in \mathbb{Z}\), is a solution.
- Divide by \(\cos^n (x)\).
\[ a_0 \frac{\sin^n x}{\cos^n x} + a_1 \frac{\sin^{n-1} x}{\cos^{n-1} x} + \cdots + a_{n-1} \frac{\sin x}{\cos x} + a_n = 0 \]
\[ a_0 \tan^n x + a_1 \tan^{n-1} x + \cdots + a_{n-1} \tan x + a_n = 0 \]
- Substitute \(t = \tan x\) and solve the polynomial equation
\[ a_0 t^n + a_1 t^{n-1} + \cdots + a_{n-1} t + a_n = 0 \]
- For each solution \(t^\ast\) to the equation in Step 3, solve the elementary trigonometric equation \(\tan x = t^\ast\).
Example
TODO
Algorithm: Solving Homogeneous Trigonometric Equations (Cotangent Substitution)
We are given the following [[Homogeneous Trigonometric Equations|homogeneous trigonometric equation]].
\[ a_0 \sin^n x + a_1 \sin^{n-1} x \cos x + \cdots + a_{n-1}\sin x \cos^{n-1} x + a_n \cos^n x = 0 \]
- Check whether \(\sin^n (x) = 0\), i.e. \(x = k\pi\) for \(k \in \mathbb{Z}\), is a solution.
- Divide by \(\sin^n (x)\).
\[ a_0 + a_1 \frac{\cos x}{\sin x} + \cdots + a_{n-1} \frac{\cos^{n-1} x}{\sin^{n-1} x} + a_n \frac{\cos^n x}{\sin^n x} = 0 \]
\[ a_0 + a_1 \cot x + \cdots + a_{n-1} \cot^{n-1} x + a_n\cot^n x = 0 \]
- Substitute \(t = \cot x\) and solve the polynomial equation
\[ a_0 + a_1 t + \cdots + a_{n-1} t^{n-1} + a_n t^n = 0 \]
- For each solution \(t^\ast\) to the equation in Step 3, solve the elementary trigonometric equation \(\cot x = t^\ast\).
Example
TODO
Other Trigonometric Equations#
Algorithm: Solving Trigonometric Equations of the Form \(a \sin x + b \cos x = c\)
We are given a trigonometric equation of the following form.
\[ a \sin x + b \cos x = c \]
- Divide both sides by \(\sqrt{a^2 + b^2}\).
\[ \frac{a}{\sqrt{a^2 + b^2}}\sin x + \frac{b}{\sqrt{a^2 + b^2}}\cos x = \frac{c}{\sqrt{a^2+b^2}} \]
- Substitute \(\cos \varphi = \frac{a}{\sqrt{a^2 + b^2}}\) and \(\sin \varphi = \frac{b}{\sqrt{a^2 + b^2}}\)
\[ \cos \varphi \sin x + \sin \varphi \cos x = \frac{c}{\sqrt{a^2+b^2}} \]
- The reason we can do this is that \(\left(\frac{a}{\sqrt{a^2 + b^2}}\right)^2 +\left(\frac{b}{\sqrt{a^2 + b^2}}\right)^2 = 1\).
- Use an identity to simplify the left-hand side.
\[ \sin (x + \varphi) = \frac{c}{\sqrt{a^2+b^2}} \]
- Requirements: \(\frac{c}{\sqrt{a^2+b^2}} \in [-1;1]\)
- Solve the elementary trigonometric equation.
\[ \begin{aligned}x + \varphi &= \arcsin\frac{c}{\sqrt{a^2+b^2}} + 2k\pi \\ x + \varphi &= -\arcsin\frac{c}{\sqrt{a^2+b^2}} + (2k+1)\pi \end{aligned} \]
- Rearrange to isolate \(x\).
\[ \begin{aligned}x &= \arcsin\frac{c}{\sqrt{a^2+b^2}} -\varphi + 2k\pi \\ x &= -\arcsin\frac{c}{\sqrt{a^2+b^2}} -\varphi + (2k+1)\pi \end{aligned} \]
6.Substitute back for \(\varphi\) to obtain the final set of solutions.
- If you substitute back \(\varphi = \arccos \frac{a}{\sqrt{a^2 + b^2}}\), you obtain
\[ \begin{aligned}x &= \arcsin\frac{c}{\sqrt{a^2+b^2}} - \arccos \frac{a}{\sqrt{a^2 + b^2}} + 2k\pi \\ x &= -\arcsin\frac{c}{\sqrt{a^2+b^2}} - \arccos \frac{a}{\sqrt{a^2 + b^2}} + (2k+1)\pi \end{aligned} \]
- If you susbtitute back \(\varphi = \arcsin \frac{b}{\sqrt{a^2 + b^2}}\), you obtain
\[ \begin{aligned}x &= \arcsin\frac{c}{\sqrt{a^2+b^2}} - \arcsin \frac{b}{\sqrt{a^2 + b^2}} + 2k\pi \\ x &= -\arcsin\frac{c}{\sqrt{a^2+b^2}} - \arcsin \frac{b}{\sqrt{a^2 + b^2}} + (2k+1)\pi \end{aligned} \]
Example
TODO