Trigonometric identities

Given that, for n

show that

Figure 1 Figure 2

Figure 1 shows the central vertical cross-section of a solid wooden ornament.

Figure 2 shows the curve with equation

The region R, shown shaded in Figure 2, is bounded by the curve, the line with equation y = and the y-axis.

The ornament is modelled by the solid of revolution formed when R is rotated 360° about the y-axis. The units are centimetres.

Using algebraic integration and the result in part (a), determine, in cm³, the volume of wood needed to make the ornament, according to the model. Give your answer to 2 significant figures.

[ Solutions based entirely on calculator technology are not acceptable.]

Given that

• the density of the wood is 0.85 g/cm³

• the mass of the ornament is 6 grams

comment on the suitability of the model.

The complex number z = e^iθ, where θ is real.

Show that

where n is a positive integer.

Show that

Hence, making your reasoning clear, determine all the solutions of

cos 5 θ + 5 cos 3 θ + 12 cos θ = 0

in the interval 0 θ < 2π

The complex number z = e^iθ, where θ is real.

Show that

where n is a positive integer.

Show that

Hence, making your reasoning clear, determine all the solutions of

cos 5 θ + 5 cos 3 θ + 12 cos θ = 0

in the interval 0 θ < 2π

Given

show that

x = –4 x = 4

Figure 1 Figure 2

Figure 1 shows a solid paperweight with a flat base.

Figure 2 shows the curve with equation

where H is a positive constant and x is in radians.

The region R, shown shaded in Figure 2, is bounded by the curve, the line with equation x = –4, the line with equation x = 4 and the x-axis.

The paperweight is modelled by the solid of revolution formed when R is rotated 180° about the x-axis.

Given that the maximum height of the paperweight is 2 cm,

write down the value of H.

Using algebraic integration and the result in part (a), determine, in cm³, the volume of the paperweight, according to the model. Give your answer to 2 decimal places.

[ Solutions based entirely on calculator technology are not acceptable.]

State a limitation of the model.

Given

show that

x = –4 x = 4

Figure 1
Figure 2

Figure 1 shows a solid paperweight with a flat base.

Figure 2 shows the curve with equation

where H is a positive constant and x is in radians.

The region R, shown shaded in Figure 2, is bounded by the curve, the line with equation x = –4, the line with equation x = 4 and the x-axis.

The paperweight is modelled by the solid of revolution formed when R is rotated 180° about the x-axis.

Given that the maximum height of the paperweight is 2 cm,

write down the value of H.

Using algebraic integration and the result in part (a), determine, in cm³, the volume of the paperweight, according to the model. Give your answer to 2 decimal places.

[ Solutions based entirely on calculator technology are not acceptable.]

State a limitation of the model.

Use the substitution t = tan to prove the identity

Use the substitution t = tan to determine the exact value of

giving your answer in simplest form.

Use de Moivre’s theorem to prove that

Hence find the distinct roots of the equation

giving your answer to 3 decimal places where appropriate.

Use de Moivre’s theorem to prove that

Hence find the distinct roots of the equation

giving your answer to 3 decimal places where appropriate.