253 lines
7.6 KiB
Markdown
253 lines
7.6 KiB
Markdown
# MATH 117: Calculus 1
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## Functions
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A **function** is a rule where each input has exactly one output, which can be determined by the **vertical line test**.
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!!! definition
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- The **domain** is the set of allowable independent values.
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- The **range** is the set of allowable dependent values.
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Functions can be **composed** to apply the result of one function to another.
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$$
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(f\circ g)(x) = f(g(x))
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$$
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!!! warning
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Composition is not commutative: $f\circ g \neq g\circ f$.
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## Inverse functions
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The inverse of a function swaps the domain and range of the original function: $f^{-1}(x)$ is the inverse of $f(x)$.. It can be determined by solving for the other variable:
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$$
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\begin{align*}
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y&=mx+b \\
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y-b&=mx \\
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x&=\frac{y-b}{m}
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\end{align*}
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$$
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Because the domain and range are simply swapped, the inverse function is just the original function reflected across the line $y=x$.
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<img src="https://upload.wikimedia.org/wikipedia/commons/1/11/Inverse_Function_Graph.png" width=300>(Source: Wikimedia Commons, public domain)</img>
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If the inverse of a function is applied to the original function, the original value is returned.
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$$f^{-1}(f(x)) = x$$
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A function is **invertible** only if it is "**one-to-one**": each output must have exactly one input. This can be tested via a horizontal line test of the original function.
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If a function is not invertible, restricting the domain may allow a **partial inverse** to be defined.
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!!! example
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<img src="https://upload.wikimedia.org/wikipedia/commons/7/70/Inverse_square_graph.svg">(Source: Wikimedia Commons, public domain)</img>
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By restricting the domain to $[0,\inf]$, the **multivalued inverse function** $y=\pm\sqrt{x}$ is reduced to just the partial inverse $y=\sqrt{x}$.
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## Symmetry
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An **even function** satisfies the property that $f(x)=f(-x)$, indicating that it is unchanged by a reflection across the y-axis.
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An **odd function** satisfies the property that $-f(x)=f(-x)$, indicating that it is unchanged by a 180° rotation about the origin.
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The following properties are always true for even and odd functions:
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- even × even = even
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- odd × odd = even
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- even × odd = odd
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Functions that are symmetric (that is, both $f(x)$ and $f(-x)$ exist) can be split into an even and odd component. Where $g(x)$ is the even component and $h(x)$ is the odd component:
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$$
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\begin{align*}
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f(x) &= g(x) + h(x) \\
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g(x) &= \frac{1}{2}(f(x) + f(-x)) \\
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h(x) &= \frac{1}{2}(f(x) - f(-x))
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\end{align*}
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$$
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!!! note
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The hyperbolic sine and cosine are the even and odd components of $f(x)=e^x$.
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$$
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\cosh x = \frac{1}{2}(e^x + e^{-x}) \\
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\sinh x = \frac{1}{2}(e^x - e^{-x})
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$$
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## Piecewise functions
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A piecewise function is one that changes formulae at certain intervals. To solve piecewise functions, each of one's intervals should be considered.
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### Absolute value function
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$$
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\begin{align*}
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|x| = \begin{cases}
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x &\text{ if } x\geq 0 \\
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-x &\text{ if } x < 0
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\end{cases}
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\end{align*}
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$$
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### Signum function
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The signum function returns the sign of its argument.
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$$
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\begin{align*}
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\text{sgn}(x)=\begin{cases}
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-1 &\text{ if } x < 0 \\
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0 &\text{ if } x = 0 \\
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1 &\text{ if } x > 0
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\end{cases}
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\end{align*}
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$$
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### Ramp function
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The ramp function makes a ramp through the origin that suddenly flatlines at 0. Where $c$ is a constant:
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$$
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\begin{align*}
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r(t)=\begin{cases}
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0 &\text{ if } x \leq 0 \\
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ct &\text{ if } x > 0
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\end{cases}
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\end{align*}
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$$
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<img src="https://upload.wikimedia.org/wikipedia/commons/c/c9/Ramp_function.svg" width=700>(Source: Wikimedia Commons, public domain)</img>
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### Floor and ceiling functions
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The floor function rounds down.
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$$\lfloor x\rfloor$$
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The ceiling function rounds up.
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$$\lceil x \rceil$$
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### Fractional part function
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In a nutshell, the fractional part function:
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- returns the part **after the decimal point** if the number is positive
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- returns 1 - **the part after the decimal point** if the number is negative
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$$\text{FRACPT}(x) = x-\lfloor x\rfloor$$
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Because this function is periodic, it can be used to limit angles to the $[0, 2\pi)$ range with:
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$$f(\theta) = 2\pi\cdot\text{FRACPT}\biggr(\frac{\theta}{2\pi}\biggr)$$
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### Heaviside function
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The Heaviside function effectively returns a boolean whether the number is greater than 0.
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$$
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\begin{align*}
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H(x) = \begin{cases}
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0 &\text{ if } t < 0 \\
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1 &\text{ if } t \geq 0
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\end{cases}
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\end{align*}
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$$
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This can be used to construct other piecewise functions by enabling them with $H(x-a)$ as a factor, where $a$ is the interval.
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In a nutshell:
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- $1-H(t-a)$ lets you "turn a function off" at at $t=a$
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- $H(t-a)$ lets you "turn a function on at $t=a$
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- $H(t-a) - H(t-b)$ leaves a function on in the interval $(a, b)$
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!!! example
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TODO: example for converting piecewise to heaviside via collecting heavisides
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and vice versa
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## Periodicity
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The function $f(t)$ is periodic only if there is a repeating pattern, i.e. such that for every $x$, there is an $f(x) = f(x + nT)$, where $T$ is the period and $n$ is any integer.
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### Circular motion
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Please see [SL Physics 1#6.1 - Circular motion](/g11/sph3u7/#61-circular-motion) and its subcategory "Angular thingies" for more information.
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## Partial function decomposition (PFD)
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In order to PFD:
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1. Factor the denominator into *irreducibly* quadratic or linear terms.
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2. For each factor, create a term. Where capital letters below are constants:
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- A linear factor $Bx+C$ has a term $\frac{A}{Bx+C}$.
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- An *irreducibly* quadratic factor $Dx^2+Ex+G$ has a term $\frac{Hx+J}{Dx^2+Ex+G}$.
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- Duplicate factors have terms with denominators with that factor to the power of 1 up to the number of times the factor is present in the original.
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4. Set the two equal to each other such that the denominators can be factored out.
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5. Create systems of equations to solve for each constant.
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!!! example
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To decompose $\frac{x}{(x+1)(x^2+x+1)}$:
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$$
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\begin{align*}
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\frac{x}{(x+1)(x^2+x+1)} &= \frac{A}{x+1} + \frac{Bx+C}{x^2+x+1} \\
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&= \frac{A(x^2+x+1) + (Bx+C)(x+1)}{(x+1)(x^2+x+1)} \\
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x &= A(x^2+x+1) + (Bx+C)(x+1) \\
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0x^2 + x + 0 &= (Ax^2 + Bx^2) + (Ax + Bx + Cx) + (A + C) \\
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\\
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&\begin{cases}
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0 = A + B \\
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1 = A + B + C \\
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0 = A + C
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\end{cases}
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\\
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A &= -1 \\
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B &= 1 \\
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C &= 1 \\
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\\
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∴ \frac{x}{(x+1)(x^2+x+1)} &= -\frac{1}{x+1} + \frac{x + 1}{x^2 + x + 1}
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\end{align*}
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$$
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## Trigonometry
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1 radian represents the angle when the length of the arc of a circle is equal to the radius. Where $s$ is the arc length:
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$$\theta=\frac{s}{r}$$
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The following table indicates the special angles that should be memorised:
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| Angle (rad) | $\frac{\pi}{6}$ | $\frac{\pi}{4}$ | $\frac{\pi}{3}$ | $\frac{\pi}{2}$ | 1 |
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| --- | --- | --- | --- | --- | --- |
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| cos | 1 | $\frac{\sqrt{3}}{2}$ | $\frac{\sqrt{2}}{2}$ | $\frac{1}{2}$ | 0 |
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| sin | 0 | $\frac{1}{2}$ | $\frac{\sqrt{2}}{2}$ | $\frac{\sqrt{3}}{2}$ | 1 |
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### Identities
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The Pythagorean identity is the one behind right angle triangles:
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$$\cos^2\theta+\sin^2\theta = 1$$
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Cosine and sine can be converted between by an angle shift:
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$$
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\cos\biggr(\theta-\frac{\pi}{2}\biggr) = \sin\theta \\
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\sin\biggr(\theta-\frac{\pi}{2}\biggr) = \cos\theta
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$$
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The **angle sum identities** allow expanding out angles:
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$$
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\cos(a+b)=\cos a\cos b - \sin a\sin b \\
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\sin(a+b)=\sin a\cos b + \cos a\sin b
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$$
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Subtracting angles is equal to the conjugates of the angle sum identities.
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The **double angle identities** simplify the angle sum identity for a specific case.
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$$
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\sin2\theta = 2\sin\theta\cos\theta \\
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$$
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The **half angle formulas** are just random shit.
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$$
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1+\tan^2\theta = \sec^2\theta \\
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\cos^2\theta = \frac{1}{2}(1+\cos2\theta) \\
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\sin^2\theta = \frac{1}{2}(1-\cos2\theta)
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$$
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