3.4 Further Trigonometry

Given that  ,  where  ,  find the possible values of cos and tan .

The following triangle shows triangle ABC, with  and  .

Given that ,  find the area of the triangle.  Give your answer in the form ,   where  .

The following triangle shows triangle ABC, with  .

Given that ,  find the value of  .

Find the exact area of triangle ABC.

By finding the value of , show that triangle ABC is isosceles.

A sector of a circle, OPQ, is such that it has radius 3.4 cm and the angle at its centre, O, is   radians.

The diagram below shows the sector of a circle .

Find the area of the sector , giving your answer to 3 significant figures.

Find the area of the triangle , giving your answer to 3 significant figures.

The canopy of a parachute and the outermost connecting cords form a sector of a circle as shown in the diagram below, with the parachutist modelled as a particle at point .

The area of the sector OAB is  .

Find the length of one of the connecting cords on the parachute.

A plastic puzzle piece is in the form of a prism with a cross-section that is the sector of a circle, as shown in the diagram below.  The radius of the sector is , and the angle at the centre is .

The height of the puzzle piece is .

The circle sector OAB is shown in the diagram below.

The angle at the centre is  , and the line segments OC and BC have lengths of  and  respectively.

Additionally, is parallel to AB, so that  and .

Show that the area of the sector is 

Show that the area of the triangle is .

Given that the area of the shaded shape is  find the value of .

Solve the equation     for  

A right-angled triangle has hypotenuse 8 cm. One of its other sides is 5 cm.

Find exact values for ,  and , where is the smallest angle in the triangle.

The graph below shows the curve with equation  in the interval 

Point  has coordinates  and is the minimum point closest to the origin.
Point  is the maximum point closest to the origin. State the coordinates of .

A straight line with equation meets the graph of  at the three points  and , as shown in the diagram.
Given that point  has coordinates , use graph symmetries to determine the coordinates of  and .

Given that  find the possible values of  and the corresponding values of .

The following diagram shows triangle ABC, with  .

Given that  and that the ratio of the length of side AB to the length of side BC is , find the exact area of triangle ABC.

A sector of a circle, OPQ, is such that the angle at its centre, O, is  radians.

Given that the area of sector OPQ in is equal to the length of the arc PQ in mm, find

giving your answers correct to 3 significant figures.

The lengths of two sides in a right-angled triangle are  and , with .

Find the possible values of , and the corresponding values of  and , where  is the smallest angle in the triangle.  All your answers should be given in terms of  and .

The diagram below shows the sector of a circle OAB, with centre O. The angle at the centre of the sector, , is 1.3 radians.  The shaded region in the diagram is the segment bounded by the arc AB and the chord AB.

Given that the difference between the perimeter of sector OAB and the perimeter of  triangle OAB is 1.05 cm, find the area of the shaded region. Give your answer correct to 3 significant figures.

A games design company produces a popular game called ‘Nugatory Enterprise’. Each game set includes plastic game pieces which are in the form of a right prism with a cross-section that is the sector of a circle, as shown in the diagram below.  The angle at the centre of the sector is , and the height of the game piece is 6 mm.

The game pieces are hollow, with a top (which is a cross-section of the prism) and three sides, but no bottom.

Given that the external surface area of the game piece is , work out the interior volume of the game piece giving your answer correct to 3 significant figures. You may ignore the thickness of the top and sides in your calculations.

The diagram below shows the sector of a circle OAB with centre O. The angle at the centre of the sector, , is  radians.  Point P is a point on line segment [OB] such that , where  is a constant with . The shaded region in the diagram is the combination of triangle ABP with the region enclosed by the arc AB and the chord AB.

If the area of triangle OAP is denoted by  and the area of the shaded region is denoted by , show that

Solve the equation

in the interval .