## SUVAT equations

1. $\hspace{10pt}V=U+AT$
2. $\hspace{10pt}S=\left(\frac{U+V}{2}\right)T$
3. $\hspace{10pt}V^2=U^2+2AS$
4. $\hspace{10pt}S=UT+\frac{1}{2}AT^2$
5. $\hspace{10pt}S=VT-\frac{1}{2}AT^2$

where

Variable Description SI unit
S displacement m (metres)
U initial velocity m/s (metres per second)
V final velocity m/s (metres per second)
A acceleration 9.8 m/s/s (metres per second per second)
T total time s (seconds)

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## Trigonometric Identities

### Fundamental Formulae $\frac{\sin(\theta)}{\cos(\theta)}=\tan(\theta)$ $\cos^2(\theta)+\sin^2(\theta)=1$

### Double Angle Formulae $\sin(2\theta)=2\sin(\theta)\cos(\theta)$ $\cos(2\theta)=\cos(\theta)^2-\sin^2(\theta)$ $\cos(2\theta)=2\cos^2(\theta)-1$ $\cos(2\theta)=1-2\sin^2(\theta)$

### Compound Angle Formulae $\sin(\alpha\pm\beta)=\sin(\alpha)\cos(\beta)\pm\cos(\alpha)\sin(\beta)$ $\cos(\alpha\pm\beta)=\sin(\alpha)\sin(\beta)\mp\cos(\alpha)\cos(\beta)$ $\tan(\alpha\pm\beta)=\frac{\tan(\alpha)\pm\tan(\beta)}{1\mp\tan(\alpha)\tan(\beta)}$

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## Transformations

### y-transformations

A y-transformation affects the y coordinates of a curve. You can identify a y-transformation as changes are made outside the brackets of y=f(x).

• $f(x)\rightarrow f(x)+4$, this is a shift in y, the x coordinates are unaffected but all the y coordinates go up by 4.
• $f(x)\rightarrow f(x)-3$, this is a shift in y, the x coordinates are unaffected but all the y coordinates go down by 3.
• $f(x)\rightarrow 2f(x)$, this is a stretch in y, the x coordinates are unaffected but all the y coordinates are doubled.
• $f(x)\rightarrow -f(x)$, this is a flip in y, the x coordinates are unaffected but all the y coordinates are flipped across the x-axis.

### x-transformations

x-transformations always behave in the opposite way to what is expected. They can be identified when changes are made inside the brackets of y=f(x).

• $f(x)\rightarrow f(x+4)$, this is a shift in the x direction, the y coordinates are unaffected but all the x coordinates go to the left by 4, the opposite direction to what is expected.
• $f(x)\rightarrow f(x-3)$, this is a shift in the x direction, the y coordinates are unaffected but all the x coordinates go to the right by 3, the opposite direction to what is expected.
• $f(x)\rightarrow f(2x)$, this is a stretch in the x direction, the y coordinates are unaffected but all the x coordinates are halved, the opposite to what is expected.
• $f(x)\rightarrow f(-x)$, this is a flip in the x direction, the y coordinates are unaffected but all the x coordinates are flipped across the y-axis.

## The Equation of a Circle

The equation of the circle whose centre is at the point (a,b) and whose radius is r is given by $(x-a)^2+(y-b)^2=r^2$. ## DIFFERENTIATION FROM FIRST PRINCIPLES $f$