Figure 1 shows circle $\displaystyle C_1$ with radius 7 cm and circle $\displaystyle C_2$ with centre $\displaystyle O$ and radius 7 cm

The circles $\displaystyle C_1$ and $\displaystyle C_2$ intersect at the point $\displaystyle A$ and at the point $\displaystyle B$

The size of angle $\displaystyle AOB$ is $\displaystyle \theta$ radians

The perimeter of the region $\displaystyle R$ shown shaded in Figure 1 is $\displaystyle \frac{28\pi}{9}$ cm

(a) Find the exact value of $\displaystyle \theta$

(b) Find the area, in $\displaystyle \mathrm{cm}^2$ to 3 significant figures, of the region $\displaystyle R$

Figure 3 shows a right pyramid with vertex $\displaystyle V$ and square base, $\displaystyle ABCD$, of side 12 cm

The size of the angle between the plane $\displaystyle VAB$ and the base $\displaystyle ABCD$ is $\displaystyle 30^\circ$

(a) Show that the height of the pyramid is $\displaystyle 2\sqrt{3}$ cm

(b) Find, in cm, the exact length of $\displaystyle VA$

(c) Find, in $\displaystyle \mathrm{cm}^2$, the exact area of triangle $\displaystyle VAD$

(d) Find, in cm, the exact length of the perpendicular from $\displaystyle D$ to $\displaystyle VA$

(e) Find, in degrees to one decimal place, the size of the obtuse angle between the plane $\displaystyle VAB$ and the plane $\displaystyle VAD$

Figure 1 shows triangle $\displaystyle ABC$ where

(a) Show that $\displaystyle x$ satisfies the equation

(b) Hence find the exact area, in $\displaystyle \mathrm{cm}^2$, of triangle $\displaystyle ABC$

Give your answer in the form $\displaystyle 9(p\sqrt{q} + \sqrt{r})$ where $\displaystyle p, q$ and $\displaystyle r$ are integers.

(a) Using a formula on page 2, show that

(b) Hence, or otherwise, solve in degrees to one decimal place

Triangle $\displaystyle ABC$ is such that

(a) Find in degrees, to one decimal place, the two possible values for angle $\displaystyle CAB$

(b) Hence, find in cm, to 3 significant figures, the shortest possible length of $\displaystyle AC$

(a) Using a formula from page 2, show that

Figure 2 shows part of the curve $\displaystyle C$ with equation $\displaystyle y = \mathrm{sin} x$ and part of the curve $\displaystyle D$ with equation $\displaystyle y = \mathrm{cos} 2x$

Curve $\displaystyle C$ and curve $\displaystyle D$ intersect at the point $\displaystyle A$

(b) Use algebra to find the exact coordinates of $\displaystyle A$

The shaded region $\displaystyle R$ is rotated through $\displaystyle 360^\circ$ about the $\displaystyle x$-axis.

(c) Use algebraic integration to find the exact volume of the solid generated.

Give your answer in the form $\displaystyle \frac{a\sqrt{a}}{b}\pi$

where $\displaystyle a$ is a prime number and $\displaystyle b$ is an integer.

Solve the equation

for $\displaystyle 0^\circ \leq A \leq 360^\circ$

In triangle $\displaystyle ABC$, $\displaystyle AC = 12$ cm, $\displaystyle BC = 14$ cm, $\displaystyle AB = x$ cm and angle $\displaystyle ABC = 30^\circ$

(a) Show that $\displaystyle x = P\sqrt{3} \pm \sqrt{95}$ where $\displaystyle P$ is a prime number.

(b) Hence or otherwise, find in $\displaystyle \mathrm{cm}^2$, the difference between the two possible areas of triangle $\displaystyle ABC$

Give your answer in the form $\displaystyle m\sqrt{n}$ where $\displaystyle m$ is a prime number and $\displaystyle n$ is an integer.

(a) Show that $\displaystyle \mathrm{cos}^4 \theta - \mathrm{sin}^4 \theta = \mathrm{cos} 2\theta$

(b) Hence, or otherwise, solve the equation

Give your solutions to 2 decimal places.

(c) Using calculus, find the exact value of $\displaystyle \int_{\frac{\pi}{16}}^{\frac{\pi}{8}} (\mathrm{cos}^4 2x - \mathrm{sin}^4 2x - 8 \mathrm{sin} 4x) \mathrm{d}x$

Give your answer in the form $\displaystyle a - b\sqrt{2}$ where $\displaystyle a$ and $\displaystyle b$ are rational numbers.

A circle has centre $\displaystyle O$ and radius 14 cm

A sector $\displaystyle AOB$ of the circle has a perimeter of 49 cm

Find, in $\displaystyle \mathrm{cm}^2$, the area of sector $\displaystyle AOB$

Figure 4 shows a right pyramid $\displaystyle ABCDE$ with

• vertex $\displaystyle E$

• horizontal square base $\displaystyle ABCD$ of side 6cm

• $\displaystyle EA = EB = EC = ED = 12$ cm

• vertical height of $\displaystyle h$ cm

(a) Show that $\displaystyle h = 3\sqrt{14}$

(b) Find, in degrees to one decimal place, the size of the angle between $\displaystyle EB$ and the plane $\displaystyle ABCD$

The midpoint of $\displaystyle AE$ is $\displaystyle P$ and the midpoint of $\displaystyle BE$ is $\displaystyle Q$

(c) Find, to the nearest degree, the size of the obtuse angle between the plane $\displaystyle EPQ$ and the plane $\displaystyle PQCD$

Figure 1 shows sector $\displaystyle OABC$ of a circle with centre $\displaystyle O$ and radius 6 cm

The size of angle $\displaystyle AOC$ is $\displaystyle \theta$ radians.

The perimeter of sector $\displaystyle OABC$ is $\displaystyle (12 + \pi)$ cm

The region shown shaded in Figure 1 is bounded by the arc $\displaystyle ABC$ and the line $\displaystyle AC$

The area of this region is $\displaystyle S \mathrm{cm}^2$

Find the exact value of $\displaystyle S$

Figure 2 shows a right pyramid $\displaystyle VABCD$ with vertex $\displaystyle V$ and rectangular base $\displaystyle ABCD$

Find in degrees, to one decimal place, the size of the angle between the plane $\displaystyle CVD$ and the base $\displaystyle ABCD$

In triangle $\displaystyle ABC$, $\displaystyle AB = 2x$ cm, $\displaystyle BC = 3x$ cm and $\displaystyle AC = 4x$ cm

The area of triangle $\displaystyle ABC$ is $\displaystyle 50 \mathrm{cm}^2$

Find, to 2 decimal places, the value of $\displaystyle x$

Triangle $\displaystyle ABC$ is such that

Given that angle $\displaystyle ABC$ is obtuse,

find, in cm to one decimal place, the length of $\displaystyle AB$

Figure 1 shows a right prism $\displaystyle ABCDEF$

Triangle $\displaystyle CDE$ is a cross section of the prism.

• $\displaystyle \angle DCE = \angle ABF = 90^\circ$

• $\displaystyle \angle EDC = \angle FAB = 40^\circ$

• $\displaystyle DE = AF = 15$ cm

• $\displaystyle EF = DA = CB = 20$ cm

Find, in degrees to one decimal place, the size of the angle between the line $\displaystyle FD$ and the plane $\displaystyle DCBA$

(i) (a) Using a formula given on page 2, show that

(b) Hence, solve the equation

(ii) Using a formula given on page 2, solve, giving your solutions as exact values

Figure 1 shows triangle $\displaystyle ABC$ where

The area of triangle $\displaystyle ABC = 36 \mathrm{cm}^2$

(a) Show that $\displaystyle x = 9$

(b) Find, in cm to 3 significant figures, the length of $\displaystyle BC$

(c) Find, in degrees to one decimal place, the size of

(i) $\displaystyle \angle ABC$

(ii) $\displaystyle \angle ACB$

Figure 2 shows the sector $\displaystyle OPQ$ of a circle with centre $\displaystyle O$ and radius $\displaystyle r$ cm.

The area of the sector is $\displaystyle A \mathrm{cm}^2$

(a) Show that $\displaystyle A = \frac{r}{2}(21 - r)$

The area of the sector must be greater than or equal to $\displaystyle 27 \mathrm{cm}^2$

(b) Find the set of possible values of $\displaystyle r$

(c) Hence write down the set of possible values of $\displaystyle \theta$

Figure 1 shows the sector $\displaystyle AOB$ of a circle with centre $\displaystyle O$ and radius $\displaystyle 3r$ cm

A circle with radius $\displaystyle r$ cm touches $\displaystyle OA$ and $\displaystyle OB$ and the arc $\displaystyle AB$

Angle $\displaystyle AOB$ is $\displaystyle \theta$ radians, where $\displaystyle 0 < \theta < \frac{\pi}{2}$

(a) Find the exact value of $\displaystyle \theta$

The area of the region shown shaded in Figure 1 is $\displaystyle 8\pi$$\displaystyle \mathrm{cm}^{2}$

(b) Find the value of $\displaystyle r$

Figure 5 shows a right triangular prism $\displaystyle ABCDEF$ where $\displaystyle ABCD$ is a rectangle.

Using a formula from page 2,

Without using a calculator,

The angle between the plane $\displaystyle AEB$ and the plane $\displaystyle ABCD$ is $\displaystyle 65^\circ$

Using formulae from page 2, show that

Hence, or otherwise,

(c) solve, giving exact values in terms of $\displaystyle \pi$

(d) use algebraic integration to find the exact value of

Figure 1 shows sector $\displaystyle ROS$ of a circle with centre $\displaystyle O$ and radius 2 cm
The size of angle $\displaystyle ROS$ is $\displaystyle \theta$ radians.

The area of sector $\displaystyle ROS$ is $\displaystyle \frac{\pi}{2} \mathrm{cm}^2$

(a) Find the exact value of $\displaystyle \theta$

The perimeter of sector $\displaystyle ROS$ is $\displaystyle P \mathrm{cm}$

(b) Find the exact value of $\displaystyle P$

Figure 4 shows a right pyramid $\displaystyle ABCDV$

The base of the pyramid is a rectangle where,

The edges $\displaystyle VA$, $\displaystyle VB$, $\displaystyle VC$ and $\displaystyle VD$ are all of equal length.

The angle between $\displaystyle VA$ and $\displaystyle ABCD$ is $\displaystyle 45^\circ$

(a) Show that $\displaystyle VA = \frac{\sqrt{10}}{2}x$ cm

(b) Find in cm, the exact height of the pyramid in terms of $\displaystyle x$

Find, in degrees to one decimal place,

(c) the size of angle $\displaystyle VBA$

(d) the size of the obtuse angle between the plane $\displaystyle AVC$ and the plane $\displaystyle BVD$

Given that the volume of the pyramid is $\displaystyle 9\sqrt{5}$$\displaystyle \mathrm{cm}^{3}$

(e) find the value of $\displaystyle x$

(a) Using a formula on page 2 show that $\displaystyle \mathrm{cos} 2A = 2 \mathrm{cos}^2 A - 1$

(b) Hence show that $\displaystyle (2 \mathrm{cos}^2 A - 1)^2 = \frac{\mathrm{cos} 4A + 1}{2}$

The curve with equation $\displaystyle y = \frac{\mathrm{sin} 2x}{2} + \frac{(2 \mathrm{cos}^2 x - 1)^2}{2} + \frac{1}{8}$ has a stationary point $\displaystyle P$ in the range $\displaystyle 0 \leq x \leq \frac{\pi}{6}$

(c) Find the exact coordinates of $\displaystyle P$

(a) Using the formulae on page 2, show that

(b) Show that

(c) Solve, for $\displaystyle 0^\circ \leq x \leq 180^\circ$ the equation, $\displaystyle \frac{\mathrm{d}y}{\mathrm{d}x} = 0$

Give your answers to the nearest whole number.

Figure 3 shows a right pyramid with a horizontal square base.

$\displaystyle O$ is the point of intersection of the diagonals of the base.

The vertex $\displaystyle V$ of the pyramid is vertically above $\displaystyle O$

(a) Show that $\displaystyle VO = \frac{\sqrt{2}}{2}x \mathrm{cm}$

(b) Find, in degrees, the size of the angle $\displaystyle AVC$

(c) Find, in degrees to one decimal place, the size of the angle between the plane $\displaystyle VAB$ and the plane $\displaystyle VDC$

The volume of the pyramid is $\displaystyle 200 \mathrm{cm}^3$

Given that the volume of a pyramid $\displaystyle = \frac{1}{3} \times \text{base area} \times \mathrm{height}$

(d) Find to 3 significant figures, the value of $\displaystyle x$

(a) Using formulae on page 2, show that

(b) Show that $\displaystyle f(\theta) = \mathrm{sin} 2\theta$

(c) Solve, in radians to 3 significant figures, for $\displaystyle -\frac{\pi}{2} \leq x \leq \frac{\pi}{2}$, the equation

(d) Using calculus, find the exact value of

Figure 2 shows triangle $\displaystyle ABC$ where

(a) Find, to 3 significant figures, the value of $\displaystyle x$

(b) Find, in degrees to one decimal place, the size of

(i) angle $\displaystyle ABC$

(ii) angle $\displaystyle ACB$

The bisector of angle $\displaystyle ABC$ meets $\displaystyle AC$ at the point $\displaystyle M$

(c) Find the area, in $\displaystyle \mathrm{cm}^2$ to 3 significant figures, of triangle $\displaystyle BMC$.

Figure 1 shows a circle, centre $\displaystyle O$, with radius $\displaystyle r$ cm.

The points $\displaystyle A$, $\displaystyle P$ and $\displaystyle B$ lie on the circle.

The obtuse angle $\displaystyle AOB = \theta$ radians.

The area of the sector $\displaystyle APBO$, shown shaded, is $\displaystyle 372.4 \mathrm{cm}^2$ and the length of the arc $\displaystyle APB$ is $\displaystyle 53.2 \mathrm{cm}$.

Find, to 3 significant figures where appropriate, the value of

(i) $\displaystyle r$

(ii) $\displaystyle \theta$

Figure 4 shows a right pyramid with vertex $\displaystyle V$ and base $\displaystyle ABCDE$ which is a regular pentagon.

Find, in degrees to one decimal place, the size of the angle between the plane $\displaystyle VBC$ and the base $\displaystyle ABCDE$

In triangle $\displaystyle ABC$, $\displaystyle AB = 3x$ cm, $\displaystyle BC = 5x$ cm and $\displaystyle \angle ABC = 110^\circ$

(a) Find, in degrees to one decimal place, the size of $\displaystyle \angle BCA$

The area of triangle $\displaystyle ABC$ is $\displaystyle 24 \mathrm{cm}^2$

(b) Find, to 3 significant figures, the value of $\displaystyle x$

A logo, $\displaystyle AEBCD$, is shown shaded in Figure 2.

The straight line $\displaystyle ABC$ is the diameter of the semicircle $\displaystyle ADC$

$\displaystyle AEB$ is an arc of a circle with centre $\displaystyle O$

All angles are measured in radians.

• $\displaystyle BC = 2x$ cm

• $\displaystyle OA = OB = x$ cm

• length of arc $\displaystyle AEB = 1.8x$ cm

The perimeter of the logo is $\displaystyle P$

(a) Show that $\displaystyle P = ax(\pi + \pi \mathrm{sin} 0.9 + b)$ where $\displaystyle a$ and $\displaystyle b$ are constants to be found.

Given that $\displaystyle x = 10$ cm,

(b) find, in $\displaystyle \mathrm{cm}^2$ to 3 significant figures, the area of the logo.

Figure 2 shows part of the curve $\displaystyle S$ with equation $\displaystyle y = \left( \mathrm{cos} 3\theta + \sqrt{3} \mathrm{sin} 3\theta \right)^{\frac{1}{2}}$

where $\displaystyle m \leq \theta \leq n$

The curve $\displaystyle S$ meets the $\displaystyle x$-axis at the point with coordinates $\displaystyle (m, 0)$ and at the point with coordinates $\displaystyle (n, 0)$

(a) Find the exact value of $\displaystyle m$ and the exact value of $\displaystyle n$

The finite region $\displaystyle R$, shown shaded in Figure 2, is bounded by the curve $\displaystyle S$, and the $\displaystyle x$-axis in the region $\displaystyle m \leq \theta \leq n$

The region $\displaystyle R$ is rotated through $\displaystyle 2\pi$ radians about the $\displaystyle x$-axis.

(b) Use calculus to find the exact volume of the solid generated.

(a) Use a formula on page 2 to show that $\displaystyle \mathrm{sin}^2 A = \frac{1}{2}(1 - \mathrm{cos} 2A)$

(b) Show that $\displaystyle \mathrm{sin}^4 x + \mathrm{cos}^4 x = \frac{3 + \mathrm{cos} 4x}{4}$

(c) Hence solve, in degrees to one decimal place, the equation

Figure 2 shows a rectangle $\displaystyle ABCD$ with $\displaystyle AB=10\,\mathrm{cm}$ and $\displaystyle AD=15\,\mathrm{cm}$. $\displaystyle E$ is the point inside the rectangle such that $\displaystyle DE=13\,\mathrm{cm}$ and $\displaystyle \angle BAE=35^\circ$.

Given that $\displaystyle \angle AED$ is obtuse, find the area, in $\displaystyle \mathrm{cm}^2$ to one decimal place, of triangle $\displaystyle BCE$.

Figure 1 shows sector $\displaystyle ORS$ of a circle with centre $\displaystyle O$ and radius $\displaystyle 4\,\mathrm{cm}$. The size of angle $\displaystyle ROS$ is $\displaystyle \theta$ radians. The area of sector $\displaystyle ORS$ is $\displaystyle 2\pi\,\mathrm{cm}^2$.

(a) Find the exact value of $\displaystyle \theta$.

(b) Find the perimeter, in $\displaystyle \mathrm{cm}$ to 3 significant figures, of the sector $\displaystyle ORS$.

The point $\displaystyle T$ lies on $\displaystyle OR$ such that $\displaystyle OT:TR=1:3$. The shaded region is bounded by $\displaystyle TR$, $\displaystyle TS$ and the arc $\displaystyle RS$. The area of this region is $\displaystyle A\,\mathrm{cm}^2$.

(c) Find the exact value of $\displaystyle A$.

Given that

(a) without using a calculator, find the value of $\displaystyle a$.

Show your working clearly.

Triangle $\displaystyle PQR$ is such that

(b) Show that

Given that

and that the area of triangle $\displaystyle PQR$ is $\displaystyle A\,\mathrm{cm}^2$,

(c) find the exact value of $\displaystyle A$ in the form

where $\displaystyle p$, $\displaystyle q$, $\displaystyle r$ and $\displaystyle s$ are integers.

Using suitable results for $\displaystyle \mathrm{sin}\,(A+B)$ and $\displaystyle \mathrm{sin}\,(A-B)$,

(a) show that

Figure 4 shows a sketch of part of the curve with equation

(b) Use calculus to find the total area of the regions bounded by the curve and the $\displaystyle x$-axis between $\displaystyle x=0$ and $\displaystyle x=\frac{\pi}{2}$.

Give your answer to 3 significant figures.

Figure 1 shows two circles, $\displaystyle C_1$ and $\displaystyle C_2$, each with a radius of $\displaystyle 6\,\mathrm{cm}$.

The centre of $\displaystyle C_1$ is $\displaystyle O_1$ such that $\displaystyle O_1$ lies on $\displaystyle C_2$. The centre of $\displaystyle C_2$ is $\displaystyle O_2$ such that $\displaystyle O_2$ lies on $\displaystyle C_1$.

The circles intersect at the points $\displaystyle A$ and $\displaystyle B$ and enclose the region $\displaystyle R$, shown shaded in Figure 1.

The area of region $\displaystyle R$ is $\displaystyle P\,\mathrm{cm}^2$.

Find the exact value of $\displaystyle P$, giving your answer in the form

where $\displaystyle a$, $\displaystyle b$ and $\displaystyle c$ are integers.

(i) (a) Using a formula from page 2, show that

Given that $\displaystyle \mathrm{tan}\,2\alpha=1$,

(b) show that

where $\displaystyle a$ and $\displaystyle b$ are integers whose values need to be found.

(ii) (a) Using formulae from page 2, show that

can be written as $\displaystyle \mathrm{tan}\,x^\circ=1$.

(b) Hence, or otherwise, solve

for $\displaystyle -90<y\leq90$.

Figure 1 shows a right pyramid $\displaystyle VABCD$ with vertex $\displaystyle V$ and square base $\displaystyle ABCD$. Each of the edges of the pyramid has the same length.

Find the size, in degrees to one decimal place, of the angle between the plane $\displaystyle CVD$ and the base $\displaystyle ABCD$.

Solve

(a) $$\mathrm{cos}\,(3x-15)^\circ=\frac{\sqrt{3}}{2}, \qquad 0\leq x<180,$$

(b) $$3\mathrm{tan}\,y^\circ+4\mathrm{sin}\,y^\circ=0, \qquad -180\leq y<180,$$

giving your answers to one decimal place where appropriate,

(c) $$\mathrm{cos}\,\theta^\circ=3\mathrm{sin}^2\theta^\circ-1, \qquad -180\leq\theta<180,$$

giving your answers to one decimal place where appropriate.

Angle $\displaystyle \alpha$ is acute such that $\displaystyle \mathrm{cos}\,\alpha=\frac35$.

Angle $\displaystyle \beta$ is obtuse such that $\displaystyle \mathrm{sin}\,\beta=\frac12$.

(a) Find the exact value of

(i) $\displaystyle \mathrm{tan}\,\alpha$,

(ii) $\displaystyle \mathrm{tan}\,\beta$.

(b) Hence show that

where $\displaystyle m$ and $\displaystyle n$ are positive integers whose values are to be found.

Figure 1 shows a sector $\displaystyle OPQ$ of a circle with centre $\displaystyle O$. The radius of the circle is $\displaystyle 18\,\mathrm{cm}$ and the angle $\displaystyle POQ$ is $\displaystyle \frac{2\pi}{3}$ radians.

(a) Find the length of the arc $\displaystyle PQ$, giving your answer as a multiple of $\displaystyle \pi$.

Figure 2 shows the sector $\displaystyle OPQ$ and the kite $\displaystyle OPTQ$. $\displaystyle PT$ is the tangent to the circle at $\displaystyle P$ and $\displaystyle QT$ is the tangent at $\displaystyle Q$, such that angle $\displaystyle PTQ=\alpha$ radians.

(b) (i) Find $\displaystyle \alpha$ in terms of $\displaystyle \pi$.

(ii) Calculate, to 3 significant figures, the area of the region, shown shaded in Figure 2, which is bounded by the arc $\displaystyle PQ$ and the tangents $\displaystyle PT$ and $\displaystyle QT$.

(a) Solve the equation

Give your solutions to the nearest whole number.

Given that

(b) show that

(c) Hence solve the equation

Give your solutions to the nearest whole number.

Figure 1 shows the right pyramid $\displaystyle ABCDE$. Its base $\displaystyle ABCD$ is a horizontal rectangle with $\displaystyle AD=12\,\mathrm{cm}$ and $\displaystyle CD=16\,\mathrm{cm}$. The height $\displaystyle ME$ is $\displaystyle 14\,\mathrm{cm}$, where $\displaystyle M$ is the point of intersection of the diagonals of the base. The sloping edges are all of equal length.

(a) Calculate, to 3 significant figures, the length of a sloping edge.

Calculate, in degrees to one decimal place, the size of

(b) the angle between $\displaystyle AE$ and the base,

(c) the angle between the plane $\displaystyle AED$ and the base.

Figure 3 shows triangle $\displaystyle ABC$ with $\displaystyle AB=12\,\mathrm{cm}$, $\displaystyle BC=6\,\mathrm{cm}$ and $\displaystyle AC=2x\,\mathrm{cm}$. The point $\displaystyle D$ is the midpoint of $\displaystyle AC$ and $\displaystyle BD=6\,\mathrm{cm}$.

$\displaystyle \angle ABD=\theta^\circ$ and $\displaystyle \angle DBC=\phi^\circ$, where $\displaystyle \theta\neq0$ and $\displaystyle \phi\neq0$.

(a) Show that

(b) Hence, or otherwise, show that

(c) Show that

(d) Hence show that

Here is a formula

(a) Find the exact value of $\displaystyle P$ when $\displaystyle t=\frac{10}{3}$.

(b) Find

(i) the largest value of $\displaystyle P$,

(ii) the smallest value of $\displaystyle P$.

(c) Find the least value of $\displaystyle t$ for which $\displaystyle P=4$.

(a) Show that

(b) Hence express $\displaystyle 2\mathrm{sin}\,7x\,\mathrm{cos}\,x$ in the form

where $\displaystyle m$ and $\displaystyle n$ are integers, giving the value of $\displaystyle m$ and the value of $\displaystyle n$.

(c) Use calculus to evaluate

Solve

(a) $$\mathrm{sin}\left(x+\frac{\pi}{3}\right)=\frac{\sqrt{3}}{2}, \qquad 0\leq x\leq2\pi,$$

giving your answers in terms of $\displaystyle \pi$,

(b) $$3\mathrm{sin}\,\theta+5\mathrm{cos}\,\theta=0, \qquad -360^\circ\leq\theta\leq360^\circ,$$

giving your answers to the nearest degree,

(c) $$1+\mathrm{sin}\,2y=2\mathrm{cos}^2 2y, \qquad -180^\circ\leq y\leq0^\circ.$$

(a) Show that

(b) Hence show that