3.8 The unit circle and trigonometric equations questions for IB Maths AI

Show that ${\mathrm{e}}^{\mathrm{i}\theta }+{\mathrm{e}}^{-\mathrm{i}\theta }=2\cos \theta$.

By expanding ${({\mathrm{e}}^{\mathrm{i}\theta }+{\mathrm{e}}^{-\mathrm{i}\theta })}^2$, show that $4{\cos }^2\theta =2\cos 2\theta +2$.

Hence find the exact value of ${\cos }^2{\displaystyle \frac{\pi }{8}}$.

Write down, in radians, the angle at the centre of the wheel between two adjacent capsules. Give your answer as an exact multiple of $\pi$.

Find the distance between two adjacent capsules, measured along the rim.

Find the straight-line distance between two adjacent capsules.

Write down the angular speed of the wheel in radians per minute.

Find the distance travelled by a capsule in 4 minutes.

Valentina boards a capsule at the lowest point of the wheel at time $t=0$. Her height above the ground, $h$ metres, $t$ minutes later is modelled by

Find Valentina's height after 4 minutes.

Find the length of time, during one rotation, for which Valentina is more than 70 m above the ground. Give your answer in minutes.

Write down $V_1$ and $V_2$ each as the imaginary part of a complex function of the form $a{\mathrm{e}}^{\mathrm{i}(100\pi t+c)}$.

Show that $V_1+V_2=\mathrm{I}\mathrm{m}\left(z{\mathrm{e}}^{\mathrm{i}100\pi t}\right)$, where $z=30+40{\mathrm{e}}^{\mathrm{i}\frac{\pi }{3}}$.

Find $z$ in the form $a+b\mathrm{i}$, giving the exact values of $a$ and $b$.

Hence write $V_1+V_2$ in the form $A\sin (100\pi t+\phi )$, where $A>0$ and $0<\phi <\frac{\pi }{2}$.

Find the first time $t>0$ at which the total voltage $V_1+V_2$ is a maximum. Give your answer in milliseconds.

The total voltage is applied to a circuit in which a resistor, an inductor and a capacitor are connected in series. Each component has a complex impedance:

• a resistor of $R$ ohms has impedance $R$

• an inductor of $L$ henrys has impedance $\mathrm{i}\omega L$

• a capacitor of $C$ farads has impedance $\frac{1}{\mathrm{i}\omega C}$

where $\omega$ is the angular frequency. For components in series, the total impedance $Z$ is the sum of their impedances. In Kwame's circuit $R=20$, $L=0.15$ and $C=100\times {10}^{-6}$.

Find $Z$ in the form $a+b\mathrm{i}$.

Find the modulus and the argument of $Z$.

The current in the circuit, $I$ amperes, is given by $I=\mathrm{I}\mathrm{m}\left(w{\mathrm{e}}^{\mathrm{i}100\pi t}\right)$, where $w=\frac{z}{Z}$.

Find $I$ in the form $B\sin (100\pi t+\psi )$, where $B>0$.

Find the time interval between a maximum of the voltage and the next maximum of the current. Give your answer in milliseconds.

The circuit is said to be at resonance when its total impedance is a real number.

Kwame can change the angular frequency of the supply. Find the frequency, $f=\frac{\omega }{2\pi }$ hertz, at which this circuit is at resonance.

Kwame now uses the original supply with only the resistor and the inductor, connected in parallel. For two components in parallel, the total impedance $Z_{\text{p}}$ satisfies

Find $Z_{\text{p}}$ in the form $a+b\mathrm{i}$.

Hence find the amplitude of the current in this parallel circuit.

State one reason why the current measured by Kwame might differ from the value predicted by this model.

Find the coordinates of $P$.

The marker is now moved to the point $Q$, the reflection of $P$ in the $y$-axis. Write down the angle, in radians, between $OQ$ and the positive $x$-axis.

State whether $\tan 2.4$ is positive or negative. Give a reason for your answer.

Write down the greatest angle between the rod and the vertical, in radians. Hence find this angle in degrees.

Find the length of the arc travelled by the bob as it swings from one extreme position to the other.

Find the period of the model.

Find all the times in the first 3 seconds at which $\theta =0.15$.

The horizontal displacement of the bob from its lowest position is $x=1.2\sin \theta$ metres. Find the greatest value of $x$.

Find ${\theta }^{\mathrm{\prime }}(t)$.

The speed of the bob is $1.2\mid {\theta }^{\mathrm{\prime }}(t)\mid$ ${\text{m s}}^{-1}$. Find the greatest speed of the bob and the first time at which it occurs.

Suggest one reason why this model may not be suitable for large values of $t$.

Find the exact value of $\cos \theta$.

Hence find the exact value of $\tan \theta$.

Find the value of $\theta$.