Trigonometric Functions (AQA A Level Maths: Pure): Exam Questions

You are given that $\tan \left(-45°\right)=-1$

By sketching an appropriate graph, find all the solutions of

$\tan \theta =-1$

in the interval $0°\le \theta \le 360°$.

(i) Sketch the graph of $y=\cos \left(\theta +30°\right)$ in the interval $-180°\le \theta \le 360°$.

(ii) Use the graph to find all the values of $\theta$ for which $\cos \left(\theta +30°\right)=0$ in the given interval.

(i) Sketch the graph of $y=\sin \theta$ in the interval $0°\le \theta \le 360°$.

The sketch must include coordinates of all points where the graph meets the coordinate axes.

(ii) Given that $\sin 30°=0.5$, use your graph to find another value of $\theta$ in the given range for which $\sin \theta =0.5$.

(i) On the same set of axes, sketch the graphs of $y=\cos \theta$ and $y=\cos 3\theta$ where $-180°\le \theta \le 180°$.

Label the coordinates of all points of intersection with the coordinate axes.

(ii) State the coordinates of any points where the two graphs intersect.

The graph below shows the curve with equation $y=\tan \left(x+50°\right)$, in the interval $-180°\le x\le 360°$.

A student states that the curve could also have the equation $y=\tan \left(x-130°\right)$.

Is the student correct? Give a reason for your answer.

For the graph shown in part (a), find the coordinates of all the points of intersection between the curve and the coordinate axes within the given interval.

Find a different example of an equation that represents the same curve, $y=\tan \left(x+50°\right)$.

The graph below shows the curve with equation $y=\sin 2x$ in the interval $-60°\le x\le 270°$.

Point A has coordinates $\left(-45,-1\right)$ and is the minimum point closest to the origin.

Point B is the maximum point closest to the origin.

State the coordinates of B.

The straight line with equation $y=-\frac{1}{2}$ meets the graph of $y=\sin 2x$ at the three points P, Q and R, as shown in the diagram.

Given that point P has coordinates $\left(-15,-\frac{1}{2}\right)$ , use the graph to find the coordinates of Q and R.

On the same set of axes, sketch the following curves:

(i) $y=2\sin \theta$ where $0°\le \theta \le 360°$

(ii) $y=-2\sin \theta$ where  $0°\le \theta \le 360°$

The sketch must include the coordinates of all points where the curve meets the coordinate axes.

State also the period of each curve.

(i) Describe geometrically the transformation that maps the graph of $y=\sin x$onto the graph of $y=\sin 3x$.

(ii)  On the graph of $y=\sin x$, the point Q has coordinates $\left(30,\frac{\sqrt{3}}{2}\right)$, where $x$ is in degrees. State the coordinates of the image of the point Q on the graph of $y=\sin 3x$.

A section of a new rollercoaster has a series of rises and falls. The vertical displacement of the rollercoaster carriage, $y$, measured in metres relative to a fixed reference height, can be modelled using the function

$y=30\cos \left(24t\right)°$

where $t$ is the time in seconds.

(i) Sketch the function for the interval $0\le t\le 30$.

(ii) How many times will the rollercoaster carriage fall during these 30 seconds?

(iii) How long does the model suggest it will take for the rollercoaster carriage to reach the bottom of the first fall?

Given that for all values of $k$, where $0<\left|k\right|<1$, the equation

$\cos \left(nx\right)=k,n\in \mathrm{\mathbb{N}}$

has exactly 4 solutions in the interval $0\le x<2\pi$, deduce the value of $n$.

Deduce the number of solutions of the equation

${\cos }^2\left(nx\right)=k^2$

in the interval $0\le x<3\pi$, justifying your answer.

Sketch the function $y=\cos \left(nx\right)$ for the interval $0\le x<4\pi$

(i) On the same set of axes, sketch the curves $y=\sin 2\theta$ and $y=\cos \left(\theta +90°\right)$ in the interval $-180°\le \theta \le 180°$.

Label the coordinates of all points of intersection with the coordinate axes.

(ii) Find the number of solutions to the equation $\sin 2\theta =\cos \left(\theta +90°\right)$ in the interval  $-180°\le \theta \le 180°$.

On the same set of axes, sketch the curves $y=-3\cos \theta$ and $y=\cos 3\theta$ in the interval $0°\le \theta \le 360°$.

Label the coordinates of all points of intersection with the coordinate axes.

In each case, state the period of the curve.

(i) Describe geometrically the transformation that maps the graph of $y=\tan x$ on to the graph of $y=\frac{1}{5}\tan x$.

(ii) On the graph of $y=\tan x$, the point Q has coordinates $\left(30,\frac{\sqrt{3}}{2}\right)$, where $x$ is in degrees. State the coordinates of the image of the point Q on the graph $y=\frac{1}{5}\tan x$.

Give your answer in surd form.

(i) Describe geometrically the transformation that maps the graph of $y=\frac{1}{3}\tan x$ on to the graph of $y=3\tan x$. 

(ii) On the graph of $y=\tan x$, the point S has coordinates $\left(60,\sqrt{3}\right)$ where $x$ is in degrees.

State the coordinates of point S after a transformation onto each of the graphs in part (i). Give your answers in surd form.

(i) Sketch the graph of $y=\sin \theta$ in the interval $-180°\le \theta \le 180°$.

(ii) Given that  $\sin 60°=\frac{\sqrt{3}}{2}$, use your graph to find all values of $\theta$ in the given interval for which

$\sin \theta =-\frac{\sqrt{3}}{2}$

You are given that $\tan 30°=\frac{1}{\sqrt{3}}$.

Use a suitable graph to find all the solutions to

$\tan \theta =-\frac{1}{\sqrt{3}}$

in the interval $0°\le \theta \le 360°$.

Sketch the graph of $y=\tan \left(\theta -45°\right)$ in the interval  $-360°\le \theta \le 360°$.

Use the fact that $\tan \left(45°\right)=1$ to find all the values of $\theta$ for which

$\tan \left(\theta -45°\right)=1$

in the interval  $-360°\le \theta \le 360°$.

(i) On the same set of axes, sketch the curves  $y=\frac{1}{2}\sin \theta$ and $y=\sin \left(\theta -60°\right)$ for the interval $-180°\le \theta \le 180°$.

State the coordinates of

• any points where the curves meet the horizontal axis

• any maximum and minimum points

(ii) Show algebraically that $\theta =90$ satisfies the equation $\frac{1}{2}\sin \theta =\sin \left(\theta -60°\right)$.  Hence, use your sketch to find any other solutions to the equation

$\frac{1}{2}\sin \theta =\sin \left(\theta -60°\right)$

in the interval $-180°\le \theta \le 180°$ .

The graph below shows a curve with equation $y=\cos \left(x+k°\right)$, $-360°\le x\le 360°$ , where $k$ is a constant.

The graph passes through the point with coordinates $\left(140,-1\right)$.

A student states that there is only one possible value for $k$.

Explain why the student is incorrect and state at least two possible values for $k$.

Find the coordinates of all points where the curve meets the $x$-axis in the given interval.

The graph below shows the curve with equation $y=\sin 3x$ in the interval $-60°\le x\le 150°$.

Points A and B are the stationary points closest to the origin.

State the coordinates of A and B.

The straight line with equation $y=\frac{\sqrt{3}}{2}$ meets the graph $y=\sin 3x$ at three points, R, S, and T.

Find the coordinates of R, S, and T.

Changes in the depth of water in a small tidal estuary relative to a fixed reference depth can be modelled using the function

$y=\sin \left(22.5t\right)°$

where $y$ is measured in metres and $t$ is the time in hours.

(i) Sketch the function for the interval $0\le t\le 8$.

(ii) If $t=0$ represents 2pm, between which times will the estuary be at or above the depth of $y=\frac{1}{2}$?

A series of dips and mounds caused by underground mining has a cross-section which can be modelled using the function

$y=4\cos \left(18x\right)°$

where $x$ and $y$ are the horizontal and vertical displacements of the ground, in metres, from a fixed origin.

(i) Sketch the function for the interval $0\le x\le 40$ and state the period of the model.

(ii) How many dips are in this model in the given interval?

(i) On the same set of axes, sketch the curves $y=\cos \left(-2\theta \right)$ and $y=\cos \frac{1}{2}\theta$ in the interval $-360°\le \theta \le 360°$.

Show clearly the coordinates of any points where the curves meet the coordinate axes.

(ii) State the period of each function.

A function is given by $\mathrm{f}\left(x\right)=\cos px$, where $0°\le x\le 360°$.

The graph of $y=\mathrm{f}\left(x\right)$ first crosses the $x$-axis at $\left(18,0\right)$.

(i) Determine the value of $p$ and sketch the graph of $y=\mathrm{f}\left(x\right)$.

(ii) State the period of $\mathrm{f}\left(x\right)$.

(i) On the same set of axes, sketch the curves $y=\tan \frac{1}{4}\theta$ and $y=\cos \left(\theta +120°\right)$ in the interval $0°\le \theta \le 270°$.

Show clearly the coordinates of any points where the curves meet the coordinate axes.

(ii) Find the number of solutions to the equation

$\cos \left(\theta +120°\right)-\tan \frac{1}{4}\theta =0$

in the interval $0°\le \theta \le 270°$.

On the same set of axes, sketch the graphs of $y=\sin \frac{1}{2}\theta$ and $y=\sin \left(\theta +30°\right)$ in the interval $-270°\le \theta \le 270°$.

Label the coordinates of points of intersection with the coordinate axes and of maximum and minimum points where appropriate.

Find the solution to the equation $\sin \frac{1}{2}\theta =\sin \left(\theta +30°\right)$ within the interval $-90°\le \theta \le 0°$.

Hence, determine the coordinates of the corresponding point of intersection between the two graphs in part (a).

On the same set of axes, sketch the curves $y=\tan \frac{1}{2}\theta$ and $y=\tan \left(\theta -30°\right)$ in the interval $-360°\le \theta \le 360°$.

Show clearly the coordinates of any points where the curves meet the horizontal axis.

In the interval $-360°\le \theta \le 360°$, find the coordinates of the two points on your sketch at which

$\tan \frac{1}{2}\theta =\tan \left(\theta -30°\right)$

Give your answer in surd form.

The graph below shows part of the curve with equation $y=\sin \left(x+k°\right)$, where $k$  is a constant.

A student states that there are an infinite number of possible values for $k$.

Is the student correct? You must explain your answer.

Another student claims that the curve shown could also have the equation $y=\cos \left(x+k°\right)$.

Find a value for $k$ for which this student is correct.

The graph below shows two curves with equations $y=p\sin x$ and $y=\cos \left(x+q°\right)$, in the interval $-180°\le x\le 180°$, where $p$ and $q$ are integers.

• The graph of $y=p\sin x$ passes through the point $\left(-90,-2\right)$

• The graph of $y=\cos \left(x+q°\right)$ passes through the point $\left(-150,0\right)$

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

For $-180°\le x\le 180°$, the curves intersect at the two points, $R$ and $S$ as shown in the diagram.

The coordinates of point $R$ are (9.90, 0.34), to 2 decimal places.

Use these coordinates, along with diagram in part (a), to find the coordinates of point $S$, to 2 decimal places.

Describe geometrically the transformation that maps the graph of $y=\sin \left(x+20°\right)$ on to the graph of $y=\cos \left(x+20°\right)$.

On the same set of axes, sketch $y=\sin \left(x+20°\right)$ and $y=\cos \left(x+20°\right)$.for the interval $-180°\le x\le 180°$.

Label the coordinates of any points of intersection between the two curves.