Trigonometric graphs and identitiesEdexcel A-Level Maths: Revision notes

The curves $y=\sin x$ and $y=\cos x$ are periodic with period $360^\circ$ ($2\pi$ radians) and oscillate between $-1$ and $1$. $y=\sin x$ passes through the origin; $y=\cos x$ starts at $(0,1)$ and is $y=\sin x$ translated $90^\circ$ to the left. Key points on $0\le x\le360^\circ$: $\sin x$ is $0,1,0,-1,0$ at $0,90,180,270,360$, and $\cos x$ is $1,0,-1,0,1$. Symmetries: $\sin(-x)=-\sin x$ (rotational symmetry about the origin), $\cos(-x)=\cos x$ (reflection in the $y$-axis), $\sin(180^\circ-x)=\sin x$, $\cos(360^\circ-x)=\cos x$ and $\sin(x+360^\circ)=\sin x$.

$y=\tan x$ has period $180^\circ$ ($\pi$ radians) and takes every real value, so it has no maximum or minimum. It is undefined wherever $\cos x=0$, giving vertical asymptotes at $x=\ldots,-90^\circ,90^\circ,270^\circ,\ldots$ The curve passes through $(0,0)$, $(180^\circ,0)$ and $(45^\circ,1)$ and is increasing between asymptotes. It has rotational symmetry about the origin: $\tan(-x)=-\tan x$.

Apply the standard rules to $y=\sin x$, $y=\cos x$ and $y=\tan x$: $y=f(x)+a$: translation $\begin{pmatrix}0\\ a\end{pmatrix}$. $y=f(x+a)$: translation $\begin{pmatrix}-a\\0\end{pmatrix}$. $y=af(x)$: stretch parallel to the $y$-axis, scale factor $a$. $y=f(ax)$: stretch parallel to the $x$-axis, scale factor $\frac1a$. Examples: $y=\sin(x+30^\circ)$ is a translation $30^\circ$ to the left, with $y$-intercept $\frac12$ and maximum at $(60^\circ,1)$. $y=\tan2x$ has period $90^\circ$ and asymptotes at $x=45^\circ$, $135^\circ,\ldots$ $y=2+3\cos(x-40^\circ)$ has maximum $5$ at $x=40^\circ$ and minimum $-1$ at $x=220^\circ$.

For all angles $\theta$ (degrees or radians): $\tan\theta=\frac{\sin\theta}{\cos\theta},\qquad\sin^2\theta+\cos^2\theta=1.$ The second comes from Pythagoras on the unit circle. Rearrangements: $\sin^2\theta=1-\cos^2\theta$ and $\cos^2\theta=1-\sin^2\theta$. An identity is true for every value of $\theta$ and uses $\equiv$. Example: if $\sin\theta=\frac35$ and $\theta$ is acute, $\cos^2\theta=\frac{16}{25}$ so $\cos\theta=\frac45$ and $\tan\theta=\frac34$.

Start with the more complicated side and work towards the other, writing each step. Do not move terms across and treat the identity as an equation. Useful moves: write $\tan\theta=\frac{\sin\theta}{\cos\theta}$, combine fractions over a common denominator, replace $1-\cos^2\theta$ by $\sin^2\theta$, and factorise to cancel. Example: $\frac{\sin^2\theta}{1-\cos\theta}=\frac{1-\cos^2\theta}{1-\cos\theta}=\frac{(1-\cos\theta)(1+\cos\theta)}{1-\cos\theta}=1+\cos\theta$. Another: $\tan\theta+\frac{1}{\tan\theta}=\frac{\sin^2\theta+\cos^2\theta}{\sin\theta\cos\theta}=\frac{1}{\sin\theta\cos\theta}$.

If an equation mixes $\sin$ and $\cos$, use $\tan\theta=\frac{\sin\theta}{\cos\theta}$ to turn $\sin x=3\cos x$ into $\tan x=3$, or use $\cos^2x=1-\sin^2x$ to turn it into a quadratic in $\sin x$. For example, $6\cos^2x+\sin x-5=0$ becomes $6\sin^2x-\sin x-1=0$. The solving step is covered in the next subtopic; here the skill is choosing the right identity.