@?further-maths

How could you rewrite the hypotenuse in terms of $\cos$ and $\sin$?? $$ \frac{\sin \theta}{\cos \theta} $$ Because of Pythagoras, what could you write?? $$ \sin^2\theta + \cos^2\theta = 1 $$ How could you simplify $7(1 - \cos^2\theta)$?? $$ 7\sin^2\theta $$ How could you rewrite $(\cos^4\theta - \sin^4\theta)$ as the difference of two squares?? $$ (\cos^2\theta - \sin^2\theta)(\cos^2\theta + \sin^2\theta) $$ How could you simplify $(\cos^2\theta - \sin^2\theta)(\cos^2\theta + \sin^2\theta)$?? $$ (\cos^2\theta - \sin^2\theta) $$

What is a linear transformation?? A transformation that only involves linear terms in $x$ and $y$. $$\left(\begin{matrix}x \ y\end{matrix}\right) \mapsto \left(\begin{matrix}x + y \ y - 1\end{matrix}\right)$$ Is that a linear transformation?? No because it’s a translation. $$\left(\begin{matrix}x \ y\end{matrix}\right) \mapsto \left(\begin{matrix}2x - y \ x + y\end{matrix}\right)$$ Is that a linear transformation?? Yes. $$\left(\begin{matrix}x \ y\end{matrix}\right) \mapsto \left(\begin{matrix}2y \ -x^2\end{matrix}\right)$$ Is that a linear transformation?? No because it involves $x^2$ terms. If something is a linear transformation of $$\left(\begin{matrix}x \ y\end{matrix}\right)$$, how could you write the resulting matrix?

See also [[Further Maths - Loci in the Argand Diagram]]S . This diagram shows the loci of points for $|z - (3 + 3i)| = 2$. Visualise the region $|z - (3 + 3i)| \le 2$?? This diagram shows the (poorly drawn) loci of points for $|z| = 3$ and $|z| = 5$. Visualise the region $3 \le |z| \le 5$?? This diagram shows the loci of points for $|z + 6| = |z - 4|$.

In , how could you write the distance between the two points?? $$ |z_2 - z_1| $$ This result is a case of what result for vectors?? $$ \vec{AB} = b - a $$ For two complex numbers $z_1$ and $z_2$, how could you write the distance between them?? $$ |z_2 - z_1| $$ If $z$ is a variable representing any complex number and $z_1$ is a fixed point, what is $|z - z_2|?? The distance from $z$ to $z_2$.

What is the definition of an even function?? $$ f(x) = f(-x) $$ What is the definition of an odd function?? $$ -f(x) = f(-x) $$ What is the parity of $x^2$?? Even What is the parity of $x^3$?? Odd What is special geometrically about even functions?? The function is symmetric with the $y$-axis. What is special geometrically about odd functions?? The function has $180^{\circ}$ rotational symmetry with respect to the origin. What is the parity of $\sin x$?

Degrees What is $\sin 0^{\circ}$?? $$ 0 $$ What is $\sin 30^{\circ}$?? $$ \frac{1}{2} $$ What is $\sin 45^{\circ}$?? $$ \frac{\sqrt{2}}{2} $$ What is $\sin 60^{\circ}$?? $$ \frac{\sqrt{3}}{2} $$ What is $\sin 90^{\circ}$?? $$ 1 $$ What is $\cos 0^{\circ}$?? $$ 1 $$ What is $\cos 30^{\circ}$?? $$ \frac{\sqrt{3}}{2} $$ What is $\cos 45^{\circ}$?? $$ \frac{\sqrt{2}}{2} $$ What is $\cos 60^{\circ}$?? $$ \frac{1}{2} $$ What is $\cos 90^{\circ}$?? $$ 0 $$ What is $\tan 0^{\circ}$?? $$ 0 $$

What is the name for $|z|$?? The modulus of $z$. What is the value of $|z|$, where $z = a + bi$?? $$ \sqrt{a^2 + b^2} $$ What is the value of $|z|^2$, where $z = a + bi$?? $$ a^2 + b^2 $$ If $|z|^2 = a^2 + b^2$, how could you also write $|z|^2$?? $$ (a + bi)(a - bi) $$ What is the word definition of the modulus of $z$?? The distance to $z$ from the origin.

Summary In summary, what is the roots of polynomials topic about?? Exploring the underlying relationship between the roots of a polynomial and the coefficient of each term. Basic Notations What are the two roots of a quadratic called?? $$ \alpha, \beta $$ What are the three roots of a cubic called?? $$ \alpha, \beta, \gamma $$ What are the four roots of a quartic called?? $$ \alpha, \beta, \gamma, \delta $$ Definitions What is the sum of the roots $\alpha + \beta$ equal to for a quadratic?

How many radians in a full circle?? $$ 2\pi $$ How many radians in $360^{\circ}$?? $$ 2\pi $$ How many circles in a half turn?? $$ \pi $$ What is $180^{\circ}$ in radians?? $$ \pi $$ What is $\frac{\pi}{2}$ radians in degrees?? $$ 90^{\circ} $$ What is $90^{\circ}$ degrees in radians?? $$ \frac{\pi}{2} $$ What is $\frac{\pi}{3}$ radians in degrees?? $$ 60^{\circ} $$ What is $60^{\circ}$ degrees in radians?? $$ \frac{\pi}{2} $$ What is $\frac{\pi}{6} radians in radians?

In summary, what is volumes of revolutions?? Finding volumes by integrating. Rotating the line $y = x$ around the $x$ axis creates what shape?? A cone. If you have a function $f(x)$ and an interval $[a,b]$, how can you find the volume of revolution around the $x$-axis?? $$ \int^b_a \pi f(x)^2 dx $$ If normal integration is an infinite summation of rectangles, then volumes of revolutions is an infinite summation of?? Cylinders. If you have a cylinder with radius $y$ and width $dx$, then what is the formula for the volume of that cylinder?

See Also https://www.desmos.com/calculator/oyp1sie3gb [[Maths - Modelling with Differentiation]]S [[Maths - Sketching Gradient Functions]]S [[Maths - Chain Rule]]S [[Maths - Product Rule]]S Flashcards What is the derivative of $x^n$?? $$ nx^{(n-1)} $$ What is the derivative of $ax^n$?? $$ anx^{(n-1)} $$ What is $f'(x)$ where $f(x) = 4x^2$?? $$ 8x $$ What is $\frac{dy}{dx}$ for $y = \frac{1}{2} x^{-4}$?? $$ -2x^{-5} $$ What is the derivative of $x^{\frac{1}{2}}$?

Other than matricies, how can you solve a system of three equations?? Using the triangle method. How does the triangle method of solving a system of three equations work?? Using back substitution. What are the steps involved in the triangle method?? Add/subtract two equations of three variables to form an equation of two variables Do this for another equation, forming an equation with the same two variables Solve this using simulataneous equations Substitute the new values and solve for the final variable Why is it called the “triangle” method of solving equations?