Trigonometric ratios

The triangle ABC is shown in the diagram below. The angle ABC is radians. The angle BCA is x radians.

Explain why angle BAC = – x

Use the sine rule to show that

where p and q are integers.

Hence state the exact value of x when

A circle with centre C has equation

The graph with equation

y = │ 3 x – 36 │ – 8

intersects the circle at the points A, B and D as shown in the diagram.

Point D is vertically below point C

State the coordinates of C

State the coordinates of D

The coordinates of A are (a,10)

Find the value of a

Fully justify your answer.

Find, in radians, the angle ADB

Give your answer to three significant figures.

Hence or otherwise find the length of the minor arc AB

Give your answer to three significant figures.

A new design for a company logo is to be made from two sectors of a circle, ORP and OQS, and a rhombus OSTR, as shown in the diagram below.

The points P, O and Q lie on a straight line and the angle ROS is radians.

A large copy of the logo, with PQ 5 metres, is to be put on a wall.

Show that the area of the logo, A square metres, is given by

Show that the maximum value of A occurs when

Fully justify your answer.

Find the exact maximum value of A

Without further calculation, state how your answers to parts (b)(i) and (b)(ii) would change if PQ were increased to 10 metres.

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 .

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.