The Unit Circle & Exact Values (DP IB Analysis & Approaches (AA): 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.

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 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 $p$.

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 $\mathrm{ABC}$.

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 $\mathrm{OAB}$.

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 $\cos \theta$.

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

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