Trigonometric Identities & Equations (DP IB Applications & Interpretation (AI): HL): Exam Questions

Given that $\sin \theta =\frac{3}{5}$, where $\frac{\pi }{2}<\theta <\pi$, find the value of $\cos \theta$ and the value of $\tan \theta$.

The following diagram shows triangle $\mathrm{ABC}$, with $\mathrm{AB}=3a$, $\mathrm{BC}=a$ and $\mathrm{AC}=7$.

Given that $\cos \mathrm{A}\hat{\mathrm{B}}\mathrm{C}=\frac{1}{2}$, find the area of triangle $\mathrm{ABC}$. Give your answer in the form $\frac{p\sqrt{3}}{q}$, where $p,q\in {\mathrm{\mathbb{Z}}}^{+}$.

The following diagram shows triangle $\mathrm{ABC}$, with $\mathrm{AB}=15$, $\mathrm{AC}=20$ and $\mathrm{BC}=x$.

Given that $\cos \mathrm{B}\hat{\mathrm{A}}\mathrm{C}=\frac{2}{3}$, find the value of $\sin \mathrm{B}\hat{\mathrm{A}}\mathrm{C}$.

Find the exact area of triangle $\mathrm{ABC}$.

By finding the value of $x$, show that triangle $\mathrm{ABC}$ is isosceles.

The diagram below shows the sector of a circle $\mathrm{OAB}$.

(i) Find the area of the sector $\mathrm{OAB}$, giving your answer to 3 significant figures.

(ii) Find the area of the triangle $\mathrm{OAB}$, giving your answer to 3 significant figures.

(iii) Find the area of the shaded segment, giving your answer to 3 significant figures.

(i) Find the length of the arc $\mathrm{AB}$.

(ii) Find the perimeter of the sector $\mathrm{OAB}$.

The canopy of a parachute and the outermost connecting cords form a sector of a circle, $\mathrm{OAB}$, as shown in the following diagram, with the parachutist modelled as a particle at point $\mathrm{O}$.

The angle $\mathrm{A}\hat{\mathrm{O}}\mathrm{B}=\frac{\pi }{4}$ radians and the area of sector $\mathrm{OAB}$ is $\frac{81\pi }{200}$ m².

Find the length of one of the connecting cords.

A plastic puzzle piece is in the form of a prism with a cross-section that is a sector of a circle, as shown in the following diagram. The radius of the sector is 8 cm and the angle at the centre is 1.2 radians.

The height of the puzzle piece is 2 cm.

(i) Find the area of the cross-section.

(ii) Hence find the volume of the puzzle piece.

The following diagram shows the sector $\mathrm{OAB}$ of a circle with centre $\mathrm{O}$, where $\mathrm{A}\hat{\mathrm{O}}\mathrm{B}=\frac{\pi }{3}$ radians.

Points $\mathrm{C}$ and $\mathrm{D}$ lie on $\left[\mathrm{OB}\right]$ and $\left[\mathrm{OA}\right]$ respectively, such that $\mathrm{OC}=\mathrm{OD}=8$ cm and $\mathrm{BC}=\mathrm{AD}=p$ cm.

Show that the area of sector $\mathrm{OAB}$ is $\frac{\pi }{6}{\left(p+8\right)}^2$ cm².

Show that the area of triangle $\mathrm{OCD}$ is $16\sqrt{3}$ cm².

The area of the shaded region $\mathrm{ABCD}$ is $\left(\frac{50\pi }{3}-16\sqrt{3}\right)$ cm².

Find the value of $p$.

Solve the equation $2\sin x=\frac{1}{\sin x}$ for $0°\le x\le 360°$.

A right-angled triangle has hypotenuse 8 cm. One of its other sides is 5 cm.

Find exact values for $\sin \theta$, $\cos \theta$ and $\tan \theta$, where $\theta$ is the smallest angle in the triangle.

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

Point $\mathrm{A}$ has coordinates $\left(-45°,-1\right)$ and is the minimum point closest to the origin. Point $\mathrm{B}$ is the maximum point closest to the origin. State the coordinates of $\mathrm{B}$.

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

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

(i) Sketch the graph of $y=\sin \left(2\theta -60°\right)$ in the interval $-180°\le \theta \le 180°$.

(ii) Write down all the values of $\theta$ for which $\sin \left(2\theta -60°\right)=0$ in the given interval.

A graph has the equation $y=\cos 3x$ for the interval $-\frac{\pi }{3}\le x\le \frac{2\pi }{3}$.

Sketch the graph on the axes below.

A straight line with equation $y=\frac{1}{2}$ intersects the graph of $y=\cos 3x$.

(i) Sketch the line $y=\frac{1}{2}$ on to the same set of axes.

(ii) Find the coordinates of the points of intersection between $y=\cos 3x$ and $y=\frac{1}{2}$.

A games design company produces a popular game called ‘Inconsequential Endeavour’. Each game set includes solid plastic game pieces in the form of a right prism with a cross-section that is a sector of a circle, as shown in the following diagram. The angle at the centre of the sector is 1.05 radians and the height of the game piece is 4 mm.

Given that the volume of the game piece is 0.412 cm³, find the radius of the sector. Give your answer correct to 3 significant figures.

The following diagram shows the sector $\mathrm{OAB}$ of a circle with centre $\mathrm{O}$ and radius 9.8 cm, where $\mathrm{A}\hat{\mathrm{O}}\mathrm{B}=0.85$ radians.

Find the area of the shaded segment, bounded by arc $\mathrm{AB}$ and chord $\left[\mathrm{AB}\right]$. Give your answer correct to 3 significant figures.

Find the perimeter of sector $\mathrm{OAB}$.

The following diagram shows triangle $\mathrm{ABC}$, with $\mathrm{AB}=4$, $\mathrm{BC}=5$ and $\mathrm{AC}=x$.

Given that $\cos \mathrm{A}\hat{\mathrm{B}}\mathrm{C}=\frac{3}{4}$, find the exact area of triangle $\mathrm{ABC}$.

Find the exact perimeter of triangle $\mathrm{ABC}$.

A sector $\mathrm{OPQ}$ of a circle has $\mathrm{P}\hat{\mathrm{O}}\mathrm{Q}=\frac{5\pi }{6}$ radians.

The area of sector $\mathrm{OPQ}$, in cm², is one-fifth of the length of arc $\mathrm{PQ}$, in cm.

(i) Show that the radius of the sector is 0.4 cm.

(ii) Hence find the area of sector $\mathrm{OPQ}$ and the length of arc $\mathrm{PQ}$.

Give your answers to part (ii) correct to 3 significant figures.

Complete the following table, giving exact values. In all cases the angle should be given between 0 and 360° or between 0 and $2\pi$ radians, as appropriate.

Given that $\sin \theta =-\frac{12}{13}$, find the possible values of $\cos \theta$ and the corresponding values of $\tan \theta$.

The following diagram shows triangle $\mathrm{ABC}$, with $\mathrm{BC}=8$ and $\mathrm{AC}=x$.

Given that $\tan \mathrm{A}\hat{\mathrm{C}}\mathrm{B}=\frac{5}{12}$ and that the area of triangle $\mathrm{ABC}$ is 20, find the value of $x$.

The lengths of two sides in a right-angled triangle are 9 cm and 12 cm.

Find the possible values of $\sin \theta$, and the corresponding values of $\cos \theta$ and $\tan \theta$, where $\theta$ is the smallest angle in the triangle.  All your answers should be given as exact values.

The following diagram shows the sector $\mathrm{OAB}$ of a circle with centre $\mathrm{O}$, where $\mathrm{A}\hat{\mathrm{O}}\mathrm{B}=\frac{5\pi }{6}$ radians. Point $\mathrm{M}$ is the midpoint of $\left[\mathrm{OA}\right]$, and the shaded region is made up of triangle $\mathrm{ABM}$ and the segment bounded by arc $\mathrm{AB}$ and chord $\left[\mathrm{AB}\right]$.

Show that the ratio of the area of triangle $\mathrm{OMB}$ to the area of the shaded region is $1:\left(\frac{10\pi }{3}-1\right)$.

Given that the area of the shaded region is $\left(30\pi -9\right)$ units², find the exact area of triangle $\mathrm{OAB}$.

Use the fact that

$16x^3-12x^2-4x+3=\left(4x-3\right)\left(4x^2-1\right)$

to fully factorise $16x^3-12x^2-4x+3$.

Hence, solve the equation

$16{\sin }^{3}3\theta -12{\sin }^{2}3\theta -4\sin 3\theta +3=0$

for $0\le \theta \le \frac{\pi }{2}$. Give your answers as exact values where possible.

(i) Sketch the graph of $y=4\sin \left(\theta +45°\right)-\frac{4}{\sqrt{2}}$ in the interval $-180°\le \theta \le 360°$.

(ii) Write down all the values of $\theta$ for which $4\sin \left(\theta +45°\right)-\frac{4}{\sqrt{2}}=0$ in the given interval.

The following diagram shows the sector $\mathrm{OAB}$ of a circle with centre $\mathrm{O}$, where $\mathrm{A}\hat{\mathrm{O}}\mathrm{B}=1.3$ radians. The shaded region is the segment bounded by arc $\mathrm{AB}$ and chord $\left[\mathrm{AB}\right]$.

Given that the difference between the perimeter of sector $\mathrm{OAB}$ and the perimeter of triangle $\mathrm{OAB}$ is 1.05 cm, find the area of the shaded region. Give your answer correct to 3 significant figures.

A games design company produces a popular game called ‘Nugatory Enterprise’. Each game set includes plastic game pieces in the form of a right prism with a cross-section that is a sector of a circle, as shown in the following diagram. The angle at the centre of the sector is 0.9 radians and the height of the game piece is 6 mm.

The game pieces are hollow, with a top (which is a cross-section of the prism) and three sides, but no bottom.

Given that the external surface area of the game piece is 4.26 cm², find the interior volume of the game piece. Give your answer correct to 3 significant figures. You may ignore the thickness of the top and sides in your calculations.

In each of the following, $\theta$ is an angle measured in radians such that $0<\theta <\frac{\pi }{2}$.

Given that $\sin \theta =p$, write down expressions for $\sin (\pi -\theta )$, $\cos (\pi -\theta )$ and $\tan (\pi -\theta )$.

Given that $\cos \theta =q$, write down expressions for $\sin (\pi +\theta )$, $\cos (\pi +\theta )$ and $\tan (\pi +\theta )$.

Given that $\tan \theta =r$, write down expressions for $\sin (2\pi -\theta )$, $\cos (2\pi -\theta )$ and $\tan (2\pi -\theta )$.

Given that $\tan \theta =-\frac{4}{3}$, find the possible values of $\sin \theta$ and the corresponding values of $\cos \theta$.

A sector $\mathrm{OPQ}$ of a circle has $\mathrm{P}\hat{\mathrm{O}}\mathrm{Q}=\frac{11\pi }{14}$ radians.

The area of sector $\mathrm{OPQ}$, in cm², is equal to the length of arc $\mathrm{PQ}$, in mm.

Find

(i) the area of sector $\mathrm{OPQ}$, in cm²

(ii) the length of arc $\mathrm{PQ}$, in cm.

Give your answers correct to 3 significant figures.

The following diagram shows triangle $\mathrm{ABC}$, with $\mathrm{AC}=\sqrt{41}$.

Given that $\tan \mathrm{A}\hat{\mathrm{B}}\mathrm{C}=-\frac{12}{5}$ and that the ratio of the length of side $\mathrm{AB}$ to the length of side $\mathrm{BC}$ is $10:13$, find the exact area of triangle $\mathrm{ABC}$.

The lengths of two sides in a right-angled triangle are $x$ and $y$, with $\sqrt{2}x<y$.

Find the possible values of $\sin \theta$, and the corresponding values of $\cos \theta$ and $\tan \theta$, where $\theta$ is the smallest angle in the triangle.  All your answers should be given in terms of $x$ and $y$.

The following diagram shows the sector $\mathrm{OAB}$ of a circle with centre $\mathrm{O}$, where $\mathrm{A}\hat{\mathrm{O}}\mathrm{B}=\frac{\pi }{6}$ radians. Point $\mathrm{P}$ lies on $\left[\mathrm{OB}\right]$ such that $\mathrm{OP}=k\times \mathrm{OB}$, where $k$ is a constant with $0<k<1$. The shaded region is made up of triangle $\mathrm{ABP}$ and the segment bounded by arc $\mathrm{AB}$ and chord $\left[\mathrm{AB}\right]$.

The area of triangle $\mathrm{OAP}$ is denoted by $a$ and the area of the shaded region is denoted by $b$. Show that

$k=\left(\frac{a}{a+b}\right)\frac{\pi }{3}$

Given that $\mathrm{OP}=7$ cm and that the area of triangle $\mathrm{OAP}$ is one half of the area of sector $\mathrm{OAB}$, show that the exact length of chord $\left[\mathrm{AB}\right]$, in centimetres, is given by

$\mathrm{AB}=\frac{42}{\pi }\sqrt{2-\sqrt{3}}$

Solve the equation

$\sin x\tan x=\frac{1}{2\cos x}$

in the interval $-180°\le x\le 90°$.