Find the value of $b$ and of $c$.

Find the period of $f$.

Find the values of $t$ when $h_A\left(t\right)=h_B\left(t\right)$.

Write down the period of this graph

Find the coordinates of A.

For the graph of $f$, write down the amplitude.

Write down the maximum and minimum value of $v$.

Use the model to find the depth of the water 10 hours after high tide.

Find the range of $g$.

Show that $f(2\pi )=2\pi$.

Sketch the graph of $y=p\left(t\right)$ for 0 ≤ $t$ ≤ 0.02 , showing clearly the coordinates of the first maximum and the first minimum.

Show that $\frac{a}{b}=\text{tan}\theta$.

How much time is there between the first low tide and the next high tide?

Comment on your answer to part (c)(i).

$A$.

Find the value of $C$.

A saw has a toothed edge which is 300 mm long. Find the number of complete teeth on this saw.

$C$.

Find the first time when the ball’s speed is changing at a rate of 2 cm s⁻².

Find the $x$-coordinates of the points of intersection of the two graphs.

Find the total time in the interval 0 ≤ $t$ ≤ 0.02 for which $p\left(t\right)$ ≥ 3.

Find the difference in height between low tide and high tide.

Find the equation of $L$.

Find the maximum speed of the ball.

Given that $p\left(t\right)$ can be written as $p\left(t\right)=a\text{sin}\left(b\left(t-c\right)\right)+d$ where $a$, $b$, $c$, $d$ > 0, use your graph to find the values of $a$, $b$, $c$ and $d$.

Sketch the graph $y=3\text{sin}x+4\text{cos}x$, for $-2\pi ⩽x⩽2\pi$

(i)     Find the value of $c$.

(ii)     Show that $b=\frac{\pi }{6}$.

(iii)     Find the value of $a$.

Consider the graph of $g\left(x\right)=2\text{sin}px$, 0 ≤ $x$ < $2\pi$, where $p$ > 0.

Find the greatest value of $p$ such that the graph of $g$ does not intersect the line $y=-1$.

Find $\text{arctan}\frac{3}{4}$, giving the answer to 3 significant figures.

Hence prove your conjectures in part (e).

Find the coordinates of ${\text{P}}_0$ and of ${\text{P}}_1$.

The $y$-intercept of the graph of $g$ is at $(0, r)$.

Given that $g\left(x\right)\ge 7$, find the smallest value of $r$.

Find the minimum height of the ball above the ground.

For the first 2 seconds of its motion, determine the amount of time that the ball is less than $1.8+0.2\sqrt{2}$ metres above the ground.

Show that the distance between the $x$-coordinates of ${\text{P}}_k$ and ${\text{P}}_{k+1}$ is $2\pi$.

Write down two transformations that will transform the graph of $y=v\left(t\right)$ onto the graph of $y=i\left(t\right)$.

Find the values of $t$ when $h_A\left(t\right)=h_B\left(t\right)$.

Hence find the number of rotations the water wheel makes in one hour.

Use your answers from part (a) to write down the value of $A$, $B$ and $D$.

Find $k$.

Determine the rate of change of $h$ when the top of the bucket is at $\text{B}$.

Show that $\frac{b}{\sqrt{a^2+b^2}}=\text{cos}\theta$.

Find the period of $g$.

The equation $g\left(x\right)=12$ has two solutions where $\pi$ ≤ $x$ ≤ $\frac{4\pi }{3}$. Find both solutions.

For the graph of $f$, write down the period.

At the points A and B on the diagram, the gradients of the two graphs are equal.

Determine the $y$-coordinate of A on the graph of $g$.

Find the value of $p$.

Hence, find the value of $f$(6).

Find the $x$-coordinates of $\text{A}$ and $\text{B}$.

Find the exact area of the shaded region, giving your answer in the form $p\pi +q\sqrt{3}$, where $p$, $q\in \mathbb{Q}$.

Find the value of $q$;

Find the value of $b$.

Find the height of the ball above the ground when it is released.

Show that the ball takes $2$ seconds to return to its initial height above the ground for the first time.

Find $\text{A}\hat{\text{O}}\text{B}$.

Find $p_{av}$(0.007).

Calculate the number of seconds it takes for the water wheel to complete one rotation.

Find the height of point $\text{A}$ above the ground.

With reference to your graph of $y=p\left(t\right)$ explain why $p_{av}\left(T\right)$ > 0 for all $T$ > 0.

Find the value of $q$.

The range of $f$ is $k$ ≤ $f\left(x\right)$ ≤ $m$. Find $k$ and $m$.

Find the value of $p$ and the value of $q$.

(i)     Find $w$.

(ii)     Hence or otherwise, find the maximum positive rate of change of $g$.

If the observer stands directly under point $\text{C}$ for one minute, point $\text{D}$ will pass over his head $n$ times.

Find the value of $n$.

Show that $b=\frac{\mathrm{\pi }}{6}$.

Find the value of $d$.

Find the smallest possible value of $c$.

Describe these two transformations.

Find the value of $a$.

Show that $b=\frac{\mathrm{\pi }}{6}$.

Find the value of $a$.

Find the smallest possible value of $c$.

Write down the amplitude of this graph

By considering the graph of $y=5\text{sin}x+12\text{cos}x$, find the value of $A$, $B$, $C$ and $D$.

Hence, write $f\left(x\right)$ in the form $p\text{cos}\left(x+r\right)$.

Find the value of $p$;

$B$.

$D$.

Find the value of $r$.

There are two high tides on 12 December 2017. At what time does the second high tide occur?

(i)     Write down the value of $k$.

(ii)     Find $g(x)$.

Find the value of $x$ for which the functions have the greatest difference.

A Ferris wheel with diameter $110$ metres rotates at a constant speed. The lowest point on the wheel is $10$ metres above the ground, as shown on the following diagram. $\text{P}$ is a point on the wheel. The wheel starts moving with $\text{P}$ at the lowest point and completes one revolution in $20$ minutes.

The height, $h$ metres, of $\text{P}$ above the ground after $t$ minutes is given by $h\left(t\right)=a \cos \left(bt\right)+c$, where $a, b, c \in \mathrm{ℝ}$.

Find the values of $a$, $b$ and $c$.

Find the height of the water at $12:00$.

Find the height of the water at $12:00$.

Determine the number of hours, over a 24-hour period, for which the tide is higher than $5$ metres.

Find the value of $d$.

Determine the number of hours, over a 24-hour period, for which the tide is higher than $5$ metres.

A fisherman notes that the water height at nearby Folkestone harbour follows the same sinusoidal pattern as that of Dungeness harbour, with the exception that high tides (and low tides) occur $50$ minutes earlier than at Dungeness.

Find a suitable equation that may be used to model the tidal height of water at Folkestone harbour.