Compound & double angle formulae

The diagram shows a sector of a circle OAB.

The point C lies on OB such that AC is perpendicular to OB.

Angle AOB is radians.

Given the area of the triangle OAC is half the area of the sector OAB, show that

Use a suitable change of sign to show that a solution to the equation

lies in the interval given by

The Newton-Raphson method is used to find an approximate solution to the equation

Using ₁ as a first approximation for apply the Newton-Raphson method twice

Give your answer to three decimal places.

Explain how a more accurate approximation for can be found using the Newton-Raphson method.

Explain why using ₁ as a first approximation in the Newton-Raphson method does not lead to a solution for .

Given that

show that

Hence, solve

in the interval 0 2

Give your answers to two decimal places.

Solve

Give your answers to one decimal place.

A robotic arm which is attached to a flat surface at the origin O, is used to draw a graphic design.

The arm is made from two rods OP and PQ, each of length d, which are joined at P.

A pen is attached to the arm at Q.

The coordinates of the pen are controlled by adjusting the angle OPQ and the angle between OP and the x-axis.

For this particular design the pen is made to move so that the two angles are always equal to each other with 0 as shown in Figure 2.

Show that the x-coordinate of the pen can be modelled by the equation

Hence, show that

It can be shown that

State the greatest possible value of x and the corresponding value of cos

Figure 3 below shows the arm when the x-coordinate is at its greatest possible value.

Find, in terms of d, the exact distance OQ.

The curve defined by the parametric equations

is shown in Figure 1 below.

Find a Cartesian equation of the curve in the form y² f (x)

The point A lies on the curve where t a

The tangent to the curve at A is at an angle to a line through A parallel to the x-axis.

The point B has coordinates (1, 0)

The line AB is at an angle to the x-axis.

By considering the gradient of the curve, show that

Find tan in terms of a.

Show that tan 2 tan

Some students are trying to prove an identity for sin (A B).

They start by drawing two right-angled triangles ODE and OEF, as shown.

The students’ incomplete proof continues,

Let angle DOE A and angle EOF B.

In triangle OFR,

Explain why in Line 4 leads to cos A sin B in Line 5

Complete Line 4 and Line 5 to prove the identity

Explain why the argument used in part (a) only proves the identity when A and B are acute angles.

Another student claims that by replacing B with B in the identity for sin (A B) it is possible to find an identity for sin (A B).

Assuming the identity for sin (A B) is correct for all values of A and B, prove a similar result for sin (A B).