Trigonometric functions and their graphsIB MYP Maths Extended: Revision notes

For $0^\circ\le x\le360^\circ$ (we use degrees only), each graph has a standard shape. $y=\sin x$ starts at $0$, rises to $1$ at $90^\circ$, returns to $0$ at $180^\circ$, falls to $-1$ at $270^\circ$ and ends at $0$ at $360^\circ$. $y=\cos x$ is the same wave starting at the top: $1$ at $0^\circ$, $0$ at $90^\circ$, $-1$ at $180^\circ$, $0$ at $270^\circ$ and $1$ at $360^\circ$. $y=\tan x$ is different: it is $0$ at $0^\circ$, $180^\circ$ and $360^\circ$, rises steeply towards $90^\circ$ and $270^\circ$, and is undefined there. The vertical lines at $x=90^\circ$ and $x=270^\circ$ are asymptotes that the graph never touches. Its values are not limited to between $-1$ and $1$.

A wave repeats itself. The period is the horizontal length of one complete cycle. For $y=\sin x$ and $y=\cos x$ the period is $360^\circ$; for $y=\tan x$ it is $180^\circ$. The amplitude is the distance from the middle of the wave to its highest point. For $y=\sin x$ and $y=\cos x$ it is $1$, because $y$ goes from $-1$ to $1$. The amplitude is always positive. Tan has no amplitude because its values are unlimited. The midline is the horizontal line halfway between the greatest and least values: $\text{midline}=\frac{\text{greatest}+\text{least}}{2}$ and $\text{amplitude}=\frac{\text{greatest}-\text{least}}{2}$.

The numbers $a$, $b$ and $c$ change the basic sine wave:

• $a$ stretches the wave vertically: the amplitude is $|a|$. If $a<0$ the graph is reflected in the $x$-axis.
• $b$ squeezes the wave horizontally: the period is $\frac{360^\circ}{b}$, so $b$ complete cycles fit into $360^\circ$.
• $c$ moves the whole graph up by $c$, so the midline is $y=c$. Example: $y=3\sin(2x)+1$ has amplitude $3$, period $\frac{360}{2}=180^\circ$ and midline $y=1$. Its greatest value is $1+3=4$ and its least is $1-3=-2$. The same rules work for $y=a\cos(bx)+c$.

To solve $\sin x=k$, $\cos x=k$ or $\tan x=k$ for $0^\circ\le x\le360^\circ$, find the first solution with the inverse function on your calculator, then use the symmetry of the graph to find the others.

• $\sin x=k$: if the first solution is $\alpha$, the second is $180^\circ-\alpha$. For $\sin x=0.5$: $x=30^\circ$ and $150^\circ$.
• $\cos x=k$: the solutions are $\alpha$ and $360^\circ-\alpha$. For $\cos x=0.5$: $x=60^\circ$ and $300^\circ$.
• $\tan x=k$: the solutions are $\alpha$ and $\alpha+180^\circ$. For $\tan x=1$: $x=45^\circ$ and $225^\circ$. If $k$ is negative, use the positive value to get the acute angle first. For $\cos x=-0.5$ the acute angle is $60^\circ$, and the solutions are $180^\circ-60^\circ=120^\circ$ and $180^\circ+60^\circ=240^\circ$.

For an equation such as $2\sin(30t)+5=6$, first rearrange to isolate the sine: $\sin(30t)=\frac12$. Treat $30t$ as a single angle, find all its solutions in the interval for $30t$, then divide to find $t$. Because the angle is $30t$, the interval for $30t$ is larger than the interval for $t$ when $b>1$. For $0\le t\le12$, $30t$ goes from $0^\circ$ to $360^\circ$. Solutions: $30t=30^\circ$ or $150^\circ$, so $t=1$ or $t=5$. Use the graph's shape to check: the wave has $b$ cycles in $360^\circ$, so $y=\sin(2x)$ crosses a given height more often than $y=\sin x$ does.