Trigonometry

Last updated Sep 4, 2023 Edit Source

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Specialist Textbook: Cambridge Chapter 10 Specialist Textbook: Cambridge Chapter 11

• Trigonometry equations have an infinite number of solutions since there is an infinite amount of ways to represent an angle
  • e.g. 360$\degree$ = 720$\degree$

• General Formula → $y = \textcolor{#E41B17}{a}{\sin}(\textcolor{#357EC7}{b}(x - \textcolor{lightgreen}{c})) + \textcolor{gold}{d}$ / $y = \textcolor{#E41B17}{a}{\cos}(\textcolor{#357EC7}{b}(x - \textcolor{lightgreen}{c})) + \textcolor{gold}{d}$
  • Amplitude: $\textcolor{#E41B17}{a}$ (a > 0)
  • Period: $\frac{2\pi}{\textcolor{#357EC7}{b}}$
  • Range: [$\textcolor{#E41B17}{-a,}$ $\textcolor{#E41B17}{a}$]
  • Domain: $\mathbb{R}$
  • Horizontal shift by $\textcolor{lightgreen}{c}$
  • Vertical shift by $\textcolor{gold}{d}$

• Base Graph 500

  • Repeats every $180\degree$ or $\pi$ radians
  • Vertical asymptotes at $90\degree\pm 180\degree$

• General Equation: $y = \textcolor{#E41B17}{a}{\tan}(\textcolor{#357EC7}{b}(x - \textcolor{lightgreen}{c})) + \textcolor{gold}{d}$

  • Components:
    • $a$ is the steepness of the graph
    • Period: $\frac{\pi}{\textcolor{#357EC7}{b}}$
      • The base period is $\pi$
    • Horizontal shift by $\textcolor{lightgreen}{c}$
    • Vertical shift by $\textcolor{gold}{d}$
  • Important Properties:
    • The period is $\frac{\pi}{b}$
    • The range is $\mathbb{R}$
    • The asymptotes have equations $x = \frac{(2k\\ +\\ 1)\pi}{2b}$
    • The $x$-intercepts are $x$ = $\frac{k\pi}{b}$

• Identities are expressions with an $=$ sign which are always true regardless of the value of $x$ or $\theta$

• Cambridge - Circular Functions 12L
• From a graph of $\textcolor{#427fbb}{\sin(\theta)}$ and $\textcolor{red}{\cos(\theta)}$ we can observe that we can transform both graphs into each other by shifting them horizontally
  • $\sin(\frac{\pi}{2} \pm \\ \theta) = \cos(\theta)$
    • The $\sin$ graph is shifted to the left
  • $\cos(\frac{\pi}{2} +\\ \theta) = \textcolor{#00ddff}{-}\sin(\theta),$ $\cos(\frac{\pi}{2} -\\ \theta) = \sin(\theta)$
    • The $\cos$ graph is shifted to the left and is the same as the $\sin$ graph reflected over the $x$-axis ​````

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    y=\sin(x)
    y=\cos(x)|RED

• $(\sin\theta^2)$ and $(\cos\theta^2)$ can be written as $\sin^2\theta$ and $\cos^2\theta$
  • $\sin^2\theta = \sin\theta * \sin\theta$
  • Write as $\sin(x)$^$2$ on ClassPad
• Since $\sin\theta$ and $\cos\theta$ represent sides of a right angled triangle inside of a circle with radius 1, using Pythagoras we know that $\sin^2\theta+\cos^2\theta = 1$ 450
  • This is known as the Pythagorean identity
  • $\textcolor{gold}{\sin^2\theta+\cos^2\theta = 1}$
    • Can also be rewritten as:
      • $1 + \tan^2{\theta} = \sec^2{\theta}$
      • $\cot^2{\theta} + 1 = \csc^2{\theta}$

• Cambridge - Circular Functions 12M
  • Proofs in textbook
• Basic Identities:
  • $\sin(A \pm B) = \sin{A}\cos{B} \pm \cos{A}\sin{B}$
  • $\cos(A \pm B) = \cos{A}\cos{B} \mp \sin{A}\sin{B}$
  • $\tan(A \pm B) = \frac{\tan{A}\\ \pm\\ \tan{B}}{1\\ \mp\\ \tan{A}\tan{B}}$
• Simplifying $a\cos{x} + b\sin{x}$:
  • $a\cos{x} + b\sin{x} = (\sqrt{a^2 + b^2})\cos(x - \alpha)$
    • $\cos{\alpha} = \frac{a}{r}$
    • $\sin{\alpha} = \frac{b}{r}$
  • $a\cos{x} + b\sin{x} = (\sqrt{a^2 + b^2})\sin(x + \beta)$
    • $\cos{\beta} = \frac{b}{r}$
    • $\sin{\beta} = \frac{a}{r}$

Adding ’like’ graphs

Adding $y = {a_1}\sin(x)$ to $y = {a_2}\sin(x)$ equals $y = (a_1 + a_2)\sin(x)$

• This also works for ’like’ cos graphs

• $\cos{A}\cos{B} = \frac{1}{2}(\cos(A − B) + \cos(A + B))$
• $\sin{A}\sin{B} = \frac{1}{2}(\cos(A − B) − \cos(A + B))$
• $\sin{A}\cos{B} = \frac{1}{2}(\sin(A + B) + \sin(A − B))$

• These are not found in the formula sheet, you must learn to derive them from the product-to-sum identities:
  • Let $P = A + B$ and $Q = A - B$
  • Then $P + Q = 2A\\ \therefore\\ A = \frac{P + Q}{2}$
  • And also $P - Q = 2B\\ \therefore\\ B = \frac{P - Q}{2}$
• $\cos{A} + \cos{B} = 2\cos(\frac{A + B}{2})\cos(\frac{A - B}{2})$
• $\cos{A} - \cos{B} = −2\sin(\frac{A + B}{2})\sin(\frac{A - B}{2})$
• $\sin{A} + \sin{B} = 2\sin(\frac{A + B}{2})\cos(\frac{A - B}{2})$
• $\sin{A} - \sin{B} = 2\sin(\frac{A - B}{2})\cos(\frac{A + B}{2})$

• Using addition formulas, we can derive useful expressions for $\sin(2\theta)$, $\cos(2\theta)$ and $\tan(2\theta)$
  • We know $2\theta = \theta + \theta$
  • $\therefore \cos(2\theta) = \cos(\theta + \theta)$
  • Make sure you know how to convert between sine and cosine so you can utilise the double angle formula
• $\sin(2\theta) = 2\sin{\theta}\cos{\theta}$
  • $\sin(\theta + \theta) = \sin{\theta}\cos{\theta} + \cos{\theta}\sin{\theta}$
• $\cos(2\theta) = 1 - 2\sin^2\theta$
  • $\cos(\theta + \theta) = \cos{\theta}\cos{\theta} - \sin{\theta}\sin{\theta}$
  • $= \cos^2\theta - \sin^2\theta$
    • ($\cos^2\theta = 1 - \sin^2\theta$)
• $\tan(2\theta) = \frac{2\tan{\theta}}{1\\ -\\ \tan^2\theta}$
  • $\tan(\theta + \theta) = \frac{\tan{\theta}\\ +\\ \tan{\theta}}{1\\ -\\ \tan{\theta}\tan{\theta}}$

• Inverse functions need to have a set range otherwise it would not be considered a function since it would not pass the vertical line test
  • This is how scientific calculators calculate a value using an inverse trig function
• Range of inverse graphs:
  • $\cos^{-1}$ has range {$\theta\\ |\\ 0 \leq \theta \leq \pi$}
  • $\sin^{-1}$ has range {$\theta\\ |\\ -\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2}$}
  • $\tan^{-1}$ has range {$\theta\\ |\\ -\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2}$}
• General solutions to trigonometric equations:
  • For $a ∈ [−1, 1]$, the general solution of the equation $\sin x = a$ is:
    • $x = 2nπ + sin^{−1} (a)$ or $x = (2n + 1)π − sin^{−1} (a)$, where $n ∈ \mathbb{Z}$
  • For $a ∈ [−1, 1]$, the general solution of the equation $\cos x = a$ is:
    • $x = 2nπ ± cos^{−1} (a)$, where $n ∈ \mathbb{Z}$
  • For $a ∈ \mathbb{R}$, the general solution of the equation $\tan x = a$ is:
    • $x = nπ + tan^{−1} (a)$, where $n ∈ \mathbb{Z}$

• Reciprocal Functions:
  • $\sec{\theta} = \frac{1}{\cos{\theta}}$
  • $\csc{\theta} = \frac{1}{\sin{\theta}}/\mathrm{cosec}\\ {\theta} = \frac{1}{\sin{\theta}}$
  • $\cot{\theta} = \frac{\cos{\theta}}{\sin{\theta}}/\cot{\theta} = \frac{1}{\tan{\theta}}$
• Pythagorean Identities:
  • $1 + \tan^2{\theta} = \sec^2{\theta}$
  • $\cot^2{\theta} + 1 = \csc^2{\theta}$

• The relationship between $y = f(x)$ and $y = \frac{1}{f(x)}$:
  • The $x$-intercepts becomes vertical asymptote
  • Positive values from $f(x)$ remain positive and vice-versa
    • Local minimum → local maximum and vice-versa
  • As $f(x)$ gets larger, $\frac{1}{f(x)}$ gets smaller and vice-versa
    • As $f(x) → \infty{^+}$, $\frac{1}{f(x)} → 0^+$
  • Where $f(x) = 1$, $\frac{1}{f(x)} = 1$ → the two graphs intercept
• The graph of cotangent is a reflection and translation of the graph of tangent: $\cot(x) = -\tan(x - \frac{\pi}{2})$

• General Equation: $y = \textcolor{#E41B17}{a}{\sec}(\textcolor{#357EC7}{b}(x - \textcolor{lightgreen}{c})) + \textcolor{gold}{d}$
  • $\textcolor{#E41B17}{a} =$ distance from line $y = 1$ and trough/peak
  • $\textcolor{#357EC7}{b} =$ (distance between troughs/peaks) $/ \frac{\pi}{2}$