Complete the table.
Degrees | Radians | sin | cos | tan |
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Complete the table.
Degrees | Radians | sin | cos | tan |
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Given that , where , find the possible values of cos and tan .
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The following triangle shows triangle ABC, with and .
Given that , find the area of the triangle. Give your answer in the form , where .
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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.
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A sector of a circle, OPQ, is such that it has radius 3.4 cm and the angle at its centre, O, is radians.
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The diagram below shows the sector of a circle .
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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.
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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 .
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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 .
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Solve the equation for
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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.
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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 .
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Complete the following table. In all cases the values for the angle should be given between 0 and 360° or 0 and radians, as appropriate.
Degrees | Radians | sin | cos | tan |
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Given that , find the possible values of and the corresponding values of .
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The following diagram shows triangle ABC, with and .
Given that and that the area of triangle ABC is equal to 20 , find the value of .
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The following diagram shows triangle ABC, with and
Given that , find the exact area of triangle ABC.
Find the exact perimeter of triangle ABC.
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A sector of a circle, OPQ, is such that the angle at its centre, O, is radians.
The area of sector OPQ in is one-fifth of the length of the arc PQ in cm.
Give your answers in part (ii) correct to 3 significant figures.
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The lengths of two sides in a right-angled triangle are 9 cm and 12 cm.
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 as exact values.
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The diagram below shows the sector of a circle , with centre O and radius 9.8 cm. The angle at the centre of the sector, , is 0.85 radians.
Find the area of the shaded segment, bounded by arc AB and chord AB. Give your answer correct to 3 significant figures.
Find the perimeter of sector OAB.
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A games design company produces a popular game called ‘Inconsequential Endeavour’. Each game set includes solid 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 .
Given that the volume of the game piece is , work out the radius of the sector. Give your answer correct to 3 significant figures.
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The diagram below shows the sector of a circle OAB with centre O. The angle at the centre of the sector, , is radians. Point M is the midpoint of line segment OA, and the shaded region is the combination of triangle ABM with the region enclosed by the arc AB and the chord AB.
Show that the ratio of the area of triangle OMB to the area of the shaded region may be expressed as
Given that the area of the shaded region is equal to , find the exact area of triangle OAB.
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A graph has the equation for the interval .
Sketch the graph on the axes below.
A straight line with equation intersects the graph of .
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Use the fact that
to fully factorise
Use your result from part (a) to solve the equation
in the interval . You should give your answers as exact values where possible.
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In each of the following, is an angle measured in radians such that .
Given that , write down expressions for and
Given that , write down expressions for and .
Given that , write down expressions for and .
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Given that find the possible values of and the corresponding values of .
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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.
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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.
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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 .
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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.
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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.
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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
Given that and that the area of triangle OAP is one half the area of sector OAB, show that the exact length of chord AB in centimetres is given by
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Solve the equation
in the interval .
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