Basic Trigonometry (AQA A Level Maths: Pure)

A triangle ABC has side lengths AB = 5 cm, BC = 4 cm and AC = 7 cm.
Calculate the size of the angle BAC, giving your answer to the nearest degree.

A triangle PQR has side lengths PQ = 12 cm, QR = 8 cm and angle PQR = 60°.
Calculate the length of RP, giving your answer to three significant figures.

Calculate the values of  and in the triangle below. Giving your answers to 3 significant figures.

Two sides of a triangular garden are 2.8 m and 12 m long, they meet at an angle of 67°. 

Calculate the area of the garden, giving your answer to 3 significant figures.

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

A triangle ABC has sides of 4 cm, 5 cm and 6 cm respectively.

Show that, for the angle between sides A and B, 

Find the value of  for the other two angles.

Two triangles both have sides AB = 5.3 cm, BC = 6.4 cm and angle ACB = 38°.

Three cruise ships are sailing around the coast. Ship B is 0.2 km south of Ship A.
Ship C is 0.9 km from Ship B on a bearing of 027°.

Calculate the distance between Ship A and Ship C.

Three telephone masts are used to triangulate locations. Mast B is due west of Mast A. Mast C is 0.6 km from Mast A and 0.7 km from Mast B on a bearing of 136°.

Show that the bearing of Mast C from Mast A is 213°.

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  between them.

Show that one possible value for is 48.1° to three significant figures.

Show that there is another possible value that works for . You must show all your working.

Using = 48.1°. Calculate the perimeter of the field. Giving your answer to three significant figures.

Alpacas are kept in a field as shown in the diagram below. The angle between fence AB and BC is 86°.

Find the length AC, to the nearest m.

Find the angle between the fences AD and CD.

Calculate the area of the field. Giving your answer to three significant figures.

A triangle ABC has sides , cm and angle BAC =

The area of the triangle is cm².

Show that satisfies the equation 

Hence, or otherwise find the value of

Find the perimeter of the triangle, to three significant figures.

Write the ratio of the angles of the triangle, to the nearest degree. Leave your answer in simplest form.

The Save My Exams logo is made up of a blue circle with a black lightning bolt through the centre. The CEO, Jamie, is preparing a giant version of this logo for a company banner. Using a small version of the logo, he models the lightning bolt as two congruent triangles ABC and DEF with side lengths AB = DE = 12 cm, BC = EF = 15 cm and
AC = DF = 7 cm as shown in the diagram below.

Calculate the size of the angle °. Give your answer in degrees to 1 decimal place

For the banner, Jamie wants to increase the side lengths of the triangle by a scale factor of 10. Find the area the lightning bolt will take up on the banner, giving your answer in m².

A triangle ABC has side lengths AB = 10 cm and AC = 17 cm, with angle BAC being an acute angle. The area of triangle ABC is 40 cm².

Find the angle BAC correct to the nearest whole degree.

Show that the side length BC is 9.43 cm, correct to 3 significant figures. 

Find the angle BCA, giving your answer to 3 significant figures.

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

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

A triangle has side lengths 7.3 cm, 9.8 cm and 10.6 cm. Calculate the size of the largest angle.

A triangle has side lengths 3 m and 5 m with an angle between them of 45°.

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

Calculate the values of and  in the diagram below. Giving your answers to 3 significant figures.

A triangle ABC has side lengths AB = cm, BC =  cm and AC = 7 cm. The angle ABC = 60°

Find the value of , giving your answer as a simplified surd.

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

A triangle PQR has sides of 4 cm, 3 cm and 2 cm respectively.

     Find the value of for the each of the angles in the triangle.

Two triangles both have sides AB = 2.8 cm, BC = 3.5 cm and angle ACB = 43°.

Find the largest angle in each of the triangles.

 Three ships are sailing through a harbour. At all times ships must be at least 50 m apart to avoid collisions. Ship A is 127 m east of Ship B. Ship C is 98 m from Ship B on a bearing of 058°

Are all the ships following the safety guidance of staying at least 50 m apart?

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° and 3.6 km from Mast A.

A triangular field has an area of 157 m². Two of the side lengths are 13.2 m and 28.4 m. Angle  is between the two given sides

Find the two possible angles for .

Given that is acute, calculate the perimeter of the field. Giving your answer to the nearest m.

Unicorns are kept in a field as shown in the diagram below. The angle between fence AB and AD is 102°.

Find the angle between the fences BC and CD.

Unicorns need at least 1234 m² each to be happy. Calculate the maximum number of unicorns that can be happily kept in the field.

A triangle ABC has sides , and angle .

The area of the triangle is   cm².

Show that  satisfies the equation .

Hence, or otherwise, find the perimeter of the triangle. Giving your answer to 3 significant figures.

Given angle ABC is obtuse, find the ratio of the angles of the triangle, 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 with AB = CD. The company wishes to have lines AB:BC:AC in the ratio 2:4:5.

The machine which makes the logo requires the angle ABC to be input accurate to
1 decimal place.

Calculate the angle ABC that should be input into the machine.

The acceptable error range of angle CAD is 1 degree. If a logo is tested and found to be outside of this range the machine requires calibration.

Find the acceptable range for to 1 decimal place

Given that the company wishes BC to be 1 inch, use the angles correct to 1 decimal place found in parts (a) and (b) to calculate the area of the quadrilateral ABCD.

An isosceles triangle has side lengths 7.3 cm and 9.8 cm. Calculate the difference between the two possible smallest angles.

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

Calculate the size of the angle BAC to two decimal places.

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

In a triangle ABC, cm,  cm and cm, angle °.

Show that .

Given that , find the area of the triangle.

Triangle PSQ and SQR are such that PS = SQ = QR. Sides PQ = 15 cm and QR =  cm. Angle PSQ = 120°.

 Calculate the exact value of .

Calculate the area of the triangle PQR. Leaving your answer in surd form.

An artist is designing a triangular sculpture, made using three equal lengths of metal piping. When laid flat the sculpture covers 21.8 m².

Calculate the total length of metal piping needed. Giving your answer to the nearest cm.

Unicorns are kept in a field as shown in the diagram below. The angle between fence AB and AD is 92°. AB and CD are parallel.

To be happy unicorns need at least 2222 m² each. Calculate the maximum number of unicorns that can happily be kept in the field.

An emergency call is picked up by an ambulance and a police car about an 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.

If both vehicles take the shortest distance to drive to the accident who will get there first? You must show all working.

 State one assumption you have made for your answer in part (a).

A triangle ABC has sides AB =  cm, BC =  cm, angle BAC =  and angle BCA = 30°.

Given that , show that .

A triangle ABC has sides ,  and angle BAC .

The area of the triangle is cm².

Find the ratio of the angles of the triangle, to the nearest degree.

Leave your answer in simplest form.