Basic Trigonometry (Edexcel A Level Maths: Pure): Exam Questions

ABC is a triangle where

• AB has length 8.4 cm

• BC has length 5.2 cm

• angle ACB is 28°

• angle BAC is °

• AC has length cm

Find the values of  and .

Give your answers to 3 significant figures.

PQR is a triangle with

• angle PQR = 39°

• angle RPQ = 128°

• PR = 2.6 cm

Find the length of QR, correct to 3 significant figures.

Triangle ABC has side lengths

• AB = 5 cm

• BC = 4 cm

• AC = 7 cm

Find the size of the angle BAC, to the nearest degree.

PQR is a triangle where

• PQ = 12 cm

• QR = 8 cm

• angle PQR = 60°

Find the length of RP, giving your answer to 3 significant figures.

Two sides of a triangular garden are 2.8 m and 12 m long.

The two sides meet at an angle of 67°. 

Find the area of the garden, to 3 significant figures.

Find the length of the third side of the garden, to 3 significant figures.

The area of a triangle WXY is 75 cm².  

Given

• XY = 14.3 cm

• angle WXY = 104°

find the length of WX, to 3 significant figures.

Ship A, ship B and ship C are sailing around the coast.

• Ship B is 0.2 km due south of ship A

• Ship C is 0.9 km from ship B on a bearing of 027° from ship B

Find the distance between ship A and ship C, to 2 significant figures.

ABC is a triangle where

• AB = 5.3 cm

• BC = 6.4 cm

• angle ACB = 38°

(i) In the case when angle BAC is obtuse, find its value, to 3 significant figures.

(ii) In the case when angle BAC is acute, find the size of angle ABC, to 2 significant figures.

A triangular field has an area of 10m².

The field has two side lengths of 3.2 m and 8.4 m which have an angle of  between them.

Show that one possible value for is 48.1°, to 1 decimal place.

Find the other possible value of , to 1 decimal place.

Using = 48.1°, find the perimeter of the field, to 3 significant figures.

A student is calculating the angles inside a triangle with side lengths 3 cm, 4 cm and 5 cm, as shown in the diagram below.

They use the cosine rule to find  to the nearest degree.

Another student uses to calculate the same angle.

(i) Show clearly that both methods give the same answer.

(ii) Give a reason for the result in part (i).

Alpacas are kept in a field, ABCD, as shown in the diagram below.

• AB has length 128 m

• BC has length 72 m

• CD has length 120 m

• AD has length 68 m

Given that the angle between AB and BC is 86°, find the length of AC, to the nearest metre.

Find the angle between AD and CD, to the nearest degree.

Find the area of the field, to 3 significant figures.

Three telephone masts are used to triangulate locations.

• Mast B is due west of mast A

• Mast C is 0.7 km from mast B on a bearing of 136° from mast B

• Mast C is 0.6 km from mast A

Find the two possible bearings of mast C from mast A.

The SaveMyExams logo is made up of a blue circle with a black lightning bolt through the centre.

The boss of the company is preparing a giant version of this logo for a company banner. 

Using a smaller version of the logo, he models the lightning bolt as two congruent triangles ABC and DEF where

• AB = DE = 12 cm

• BC = EF = 15 cm

• AC = DF = 7 cm

as shown in the diagram below.

Angle ABC is °.

Find the size of angle ABC, to 1 decimal place.

For the banner, the boss wants to increase the side lengths of the smaller logo by a scale factor of 10.

Find the area the lightning bolt will take up on the banner, to 3 significant figures, in m².

ABC is a triangle where

• AB = 10 cm

• AC = 17 cm

• angle BAC is a acute

• the area of triangle ABC is 40 cm²

Find the size of angle BAC, to the nearest degree.

Find the length of BC, to 3 significant figures. 

Find the size of angle BCA, to 3 significant figures.

A point D is placed on the side AC such that the length of BD is equal to the length of BC.

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

A parallelogram has area 50cm²

Given

• has length 14 cm

• has length 7 cm

• angle is obtuse

find the size of angle , in degrees, to 2 decimal places.

Find the length of the diagonal , in cm, to one decimal place.

Figure 1 is a sketch showing the position of three phone masts, , and .

The masts are identical and their bases are assumed to lie in the same horizontal plane.

From mast

• mast is 8.2 km away on a bearing of 072°

• mast is 15.6 km away on a bearing of 039°

Find the distance between masts and , giving your answer in km to one decimal place.

An engineer needs to travel from mast to mast .

Give a reason why the answer to part (a) is unlikely to be an accurate value for the distance the engineer travels.

ABC is a triangle where

• AB has length cm

• AC has length cm

• angle BAC is 150°

The area of the triangle is cm².

Show that satisfies the equation

Hence, find the value of .

Find the perimeter of triangle ABC, to 3 significant figures.

Show that the sizes of the three angles in triangle ABC approximately satisfy the ratio

A triangle has side lengths of 7.3 cm, 9.8 cm and 10.6 cm.

Find the size of the largest angle in the triangle, to the nearest degree.

A triangle has two sides of length 3 m and 5 m, with an angle between them of 45°.

Find the length of the third side, giving your answer to 3 significant figures.

In the diagram below

• PQ has length 13 cm

• QR has length 5.8 cm

• angle PSQ is 102°

• angle QPR is °

• SQ has length cm

Find the values of and , to 3 significant figures.

ABC is a triangle where

• AB has length cm

• BC has length cm

• AC has length 7 cm

• angle ABC is 60°

Find the exact value of .

The area of triangle ABC can be written in the form , where  and are integers.

Find and .

ABC is a triangle where

• AB = 2.8 cm

• BC = 3.5 cm

• angle ACB = 43°

A student wants to find the largest angle in triangle ABC.

By considering whether angle BAC is acute or obtuse, find the two possible values that the largest angle could be.

Give your answers to 1 decimal place.

Ship A, ship B and ship C are sailing through a harbour.

• Ship A is currently 127 m due east of Ship B

• Ship C is currently 98 m from ship B on a bearing of 058° from ship B

Safety guidance says that, at all times, ships must be at least 50 m apart to avoid collisions.

Determine whether or not the ships are currently following the safety guidance.

Three telephone masts are used to triangulate locations.

• Mast A is due north of mast B

• Mast C is 7.2 km from mast B on a bearing of 026° from mast B

• Mast C is 3.6 km from mast A

(i) Find the two possible bearings of mast C from mast A.

(ii) Given that mast A and mast B are the farthest apart of the three masts, find the distance between mast A and mast B, to 2 significant figures.

A triangular field has an area of 157 m².

• Two of the sides have length 13.2 m and 28.4 m

• ° is the angle between these two sides

Find the possible values of .

Given that is acute, find the perimeter of the field, to the nearest metre.

Unicorns are kept in a field, ABCD, as shown in the diagram below where

• AB is a fence of length 145 m

• BC is a fence of length 98 m

• CD is a fence of length 215 m

• AD is a fence of length 114 m

Given that the angle between fence AB and fence AD is 102°, find the angle between the fence BC and fence CD, to 1 decimal place.

Unicorns need at least 1234 m² of space each to be happy.

Find the maximum number of unicorns that can be kept happily in the field.

ABC is a triangle where

• AB has length cm

• AC has length cm

• angle BAC is 30°

The area of the triangle is cm².

Show that satisfies the equation

Hence, find the perimeter of triangle ABC, to 3 significant figures.

Given that angle ABC is obtuse, find the two missing angles in triangle ABC, to the nearest degree.

A car company is making a design for a new metal logo, as shown in the image below.

• Points A, B, C and D all lie on the circumference of a circle 

• AB, AC, AD, BC, and CD are chords of the circle

• AB = CD

• AB : BC : AC is equivalent to the ratio 2:4:5

• angle CAD is °

A machine which makes the logo out of metal requires the size of angle ABC to be accurate to 1 decimal place.

Find the value of angle ABC that should be input into the machine.

The machine will accept a range of values for angle CAD, as long as they are within of the true value.

Find the acceptable range of values for angle CAD that the machine will take, to 1 decimal place.

The company wishes BC to be 1 inch in length.

Find the area of the quadrilateral ABCD in square inches, to 3 significant figures.

An artist is designing a triangular sculpture, made using three equal lengths of metal piping.

When laid flat, the sculpture encloses an area of 21.8 m².

Find the total length of metal piping needed, in metres to 2 decimal places.

Figure 1 shows a sketch of triangle with cm, cm, cm, angle and angle

(i) Show that

(ii) Hence find the value of .

Hence find the value of giving your answer to one decimal place.

Triangle ABC has side lengths AB =  cm, BC =  cm and AC = 6 cm.

Find the size of the angle BAC, to 2 decimal places.

Given that the total perimeter of the triangle is 37.8 cm, find the area of the triangle, to 3 significant figures.

ABC is a triangle where

• AB has length cm

• BC has length 10 cm

• AC has length cm

• angle ABC is °

Show that

where and are integers to be found.

Given that , find the exact area of the triangle.

Triangles PSQ and SQR are such that

• PS = SQ = QR

• PQ = 15 cm

• QR =  cm

• angle PSQ = 120°

Find the exact value of .

Find the exact area of triangle PQR.

An emergency call is picked up by both an ambulance and a police car about a road traffic accident.

The police car is 15 miles due east of the ambulance and on a bearing of 038° from the accident.

The ambulance is on a bearing of 325° from the accident.

Determine which vehicle, out of the ambulance and the police car, is nearest to the accident.

You must show your working clearly.

A traffic officer says that the vehicle nearest to the accident will get to the accident first.

State one assumption that the traffic officer has made.

ABC is a triangle where

• AB has length  cm

• BC has length  cm

• angle BAC =

• angle BCA = 30°

Given that , show that

where and are integers to be found.

ABC is a triangle where

• AB has length  cm

• AC has length  cm

• angle BAC = 150°

The area of triangle ABC is cm².

Show that the sizes of the three angles in triangle ABC approximately satisfy the ratio