The Unit Circle & Exact Values (DP IB Analysis & Approaches (AA): HL): Exam Questions

Complete the following table. In all cases the values for the angle should be given between 0 and 360° or 0 and $2\mathrm{\pi }$ 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 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 ABC is equal to 20 units², find the value of $x$.

The following diagram shows triangle 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 ABC.

Find the exact perimeter of triangle ABC.

A sector of a circle, OPQ, is such that the angle at its centre, O, is $\frac{5\pi }{6}$ radians.

The area of sector OPQ in ${\mathrm{cm}}^2$ is one-fifth of the length of the arc PQ in cm.

(i) Show that the radius of the sector is equal to 0.4 cm, and hence

(ii) find the area of sector OPQ and the length of arc PQ.

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

Wynken, Blynken and Nod are three mathematics students.  While the three are revising trigonometry, Nod sets the following problem for his two companions:

“ABC is a triangle with $\mathrm{AC}=8.1\mathrm{cm},\mathrm{BC}=9.8\mathrm{cm}$ and $\mathrm{A}\hat{\mathrm{B}}\mathrm{C}=47°$. To three significant figures, what is the size of the largest angle in the triangle?”

Wynken and Blynken set to work, and several minutes pass. “70.8 degrees,” states Wynken confidently.  “118 degrees,” insists Blynken a moment later.

Demonstrate that Wynken’s and Blynken’s responses may both be correct answers to the problem Nod has set them.

Suggest an additional piece of information that Nod could provide, that would allow his problem to have a single unique solution.

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

Find the area of the shaded segment, bounded by arc AB and chord AB. Give your answer correct to 3 significant figures.

Find the perimeter of sector OAB.

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

Given that the volume of the game piece is $0.412{\mathrm{cm}}^3$, work out the radius of the sector. Give your answer correct to 3 significant figures.

The diagram below shows the sector of a circle OAB with centre O. The angle at the centre of the sector, $\mathrm{A}\hat{\mathrm{O}}\mathrm{B}$, is $\frac{5\pi }{6}$ radians. Point M is the midpoint of line segment OA, and the shaded region is the combination of triangle ABM with the region enclosed by the arc AB and the chord AB.

Show that the ratio of the area of triangle OMB to the area of the shaded region may be expressed as

$1:\left(\frac{10\pi }{3}-1\right)$

Given that the area of the shaded region is equal to $30\pi -9{\mathrm{units}}^2$, find the exact area of triangle OAB.

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

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 $\cos \theta$.

The following diagram shows triangle 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 AB to the length of side BC is $10:13$, find the exact area of triangle ABC.

A sector of a circle, OPQ, is such that the angle at its centre, O, is $\frac{11\pi }{14}$ radians.

Given that the area of sector OPQ in ${\mathrm{cm}}^2$ is equal to the length of the arc PQ in mm, find

(i) the area of sector OPQ , and

(ii) the length of arc PQ

giving your answers correct to 3 significant figures.

Triangle ABC is such that the length of side AB is $x$ units, the length of side BC is $y$ units, and $\mathrm{B}\hat{\mathrm{A}}\mathrm{C}=\theta$ is an acute angle.

Use a diagram to show that if (and only if) $x\sin \theta <y<x$ then there are two triangles, ${\mathrm{ABC}}_1$ and ${\mathrm{ABC}}_2$, which satisfy the conditions above, where ${\mathrm{C}}_1$ and ${\mathrm{C}}_2$ are points such that ${\mathrm{AC}}_1>{\mathrm{AC}}_2$.

Given that $x\sin \theta <y<x$, let angle $\mathrm{A}\widehat{{\mathrm{C}}_1}\mathrm{B}$ be denoted by $\phi$.

(i) Write down an expression for $\phi$ in terms of $x,y$ and θ.

(ii) Write down expressions for angles $\mathrm{A}\hat{\mathrm{B}}{\mathrm{C}}_1,\mathrm{A}\widehat{{\mathrm{C}}_2}\mathrm{B}$ and $\mathrm{A}\hat{\mathrm{B}}{\mathrm{C}}_2$ in terms of θ and $\phi$.

Show that the difference between the areas of triangles ${\mathrm{ABC}}_1$ and ${\mathrm{ABC}}_2$ is equal to

$\frac{1}{2}{y}^2\sin 2\varphi$

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 diagram below shows the sector of a circle OAB, with centre O. The angle at the centre of the sector, $\mathrm{A}\hat{\mathrm{O}}\mathrm{B}$, is 1.3 radians.  The shaded region in the diagram is the segment bounded by the arc AB and the chord AB.

Given that the difference between the perimeter of sector OAB and the perimeter of  triangle 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 which are in the form of a right prism with a cross-section that is the sector of a circle, as shown in the diagram below.  The angle at the centre of the sector is $0.9\mathrm{radians}$, 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 $4.26{\mathrm{cm}}^2$, work out the interior volume of the game piece giving your answer correct to 3 significant figures. You may ignore the thickness of the top and sides in your calculations.

The diagram below shows the sector of a circle OAB with centre O. The angle at the centre of the sector, $\mathrm{A}\hat{\mathrm{O}}\mathrm{B}$, is $\frac{\pi }{6}$ radians.  Point P is a point on line segment [OB] such that $\mathrm{OP}=k\times \mathrm{OB}$, where $k$ is a constant with $0<k<1$. The shaded region in the diagram is the combination of triangle ABP with the region enclosed by the arc AB and the chord AB.

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

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