Sine & cosine rules

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 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 triangle ABC.

AB is the shortest side. The lengths of AC and BC are 6.1 cm and 8.7 cm respectively.

The size of angle ABC is 38

Find the size of the largest angle.

Give your answer to the nearest degree.

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.

A circle has equation x² y² 6x 8y 264

AB is a chord of the circle.

The angle at the centre of the circle, subtended by AB, is 0.9 radians, as shown in the diagram below.

Find the area of the minor segment shaded on the diagram.

Give your answer to three significant figures.

The diagram shows a sector of a circle OAB.

C is the midpoint of OB.

Angle AOB is radians.

Given that the area of the triangle OAC is equal to one quarter of the area of the sector OAB, show that 2 sin

Use the Newton-Raphson method with ₁ , to find ₃ as an approximation for . Give your answer correct to five decimal places.

Given that 1.89549 to five decimal places, find an estimate for the percentage error in the approximation found in part (b).