Basic Trigonometry (CIE A Level Maths: Pure 1)

Exam Questions

4 hours43 questions
1
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2 marks

ABC is a right-angled triangle. AB = 38 m and angle CAB = 74°.q1-basic-trigonometry-edexcel-a-level-pure-maths-easy

Calculate the length of AC.

Give your answer correct to one decimal place.

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2
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2 marks

Find the length of the side PQ in the triangle PQR below, giving your answer to one decimal place.q2-5-1-basic-trigonometry-edexcel-a-level-pure-maths-easy

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3
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2 marks

ABC is a right-angled triangle. BC = 15 cm, AB = 7 cm and angle BAC = 90°.

A5De9tjn_q3-5-1-basic-trigonometry-edexcel-a-level-pure-maths-easy

Calculate the size of angle ACB.

Give your answer correct to three significant figures.

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4a
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3 marks

A ladder is placed against a wall. The base of the ladder is 1.2 m away from the base of the wall and it reaches 6 m up the wall.q4a-5-1-basic-trigonometry-edexcel-a-level-pure-maths-easy

To be safe to climb, the angle between the ladder and the ground must be between 65° and 75°.

Is the ladder safe to climb? You must show your working.

4b
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3 marks

Calculate the length of the ladder, give your answer to the nearest cm.

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5
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3 marks

PQR is a triangle with angle PQR = 84° and side lengths PQ = 7.5 cm and QR = 9.2 cm.

Use cosine rule to calculate the length of PR.q5-5-1-basic-trigonometry-edexcel-a-level-pure-maths-easy

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6
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3 marks

 q6-5-1-basic-trigonometry-edexcel-a-level-pure-maths-easy
x
is an acute angle. Use sine rule to calculate the angle x to the power of o. Give your answer to the nearest degree.

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7
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3 marks

A triangular field is shown in the diagram below. Calculate the area of the field, give your answer to the nearest square metre.q7-5-1-basic-trigonometry-edexcel-a-level-pure-maths-easy

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8
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3 marks

PQR is a triangle with angle PQR = 39° and angle RPQ = 128°. PR = 2.6 cm.

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

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9
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5 marks

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

She uses cosine rule to find x to the power of o to the nearest degree.

q9-5-1-basic-trigonometry-edexcel-a-level-pure-maths-easy
Another student uses SOH CAH TOA to calculate the same angle.

(i)
Show clearly how both students can achieved the same answer using either method.

(ii)
State which is the most efficient method in this case and why.

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10
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3 marks

The area of a triangle WXY is 75 cm2.  XY = 14.3 cm and angle WXY = 104°.

Using the formula A equals 1 half space a b space sin space C, calculate the length of WX. Give your answer to three significant figures.

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11
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3 marks

Show that the cosine formula a squared equals b squared plus c squared minus 2 b c space cos space A can  be rearranged into the form cos space A space equals space fraction numerator b squared plus c squared minus a squared over denominator 2 b c end fraction

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1a
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2 marks

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.

1b
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2 marks

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.

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2
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4 marks

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

q2-5-1-basic-trigonometry-edexcel-a-level-pure-maths-medium

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3a
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2 marks

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.

3b
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2 marks

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

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4
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3 marks

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

(i)
Show that, for the angle between sides A and B,cos space theta equals begin inline style 1 over 8 end style 

(ii)
Find the value of cos italic space theta  for the other two angles.

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5
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4 marks

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

(i)
Show that the angle BAC for one of the triangles is 132°, to 3 s.f.

(ii)
Find the angle ABC for the other triangle.

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6
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3 marks

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°.

q-6-5-1-basic-trigonometry-edexcel-a-level-pure-maths-medium

Calculate the distance between Ship A and Ship C.

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7
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3 marks

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°.

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8a
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3 marks

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 theta between them.

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

8b
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2 marks

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

8c
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3 marks

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

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9a
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3 marks

Alpacas are kept in a field as shown in the diagram below. The angle between fence AB and BC is 86°.q9a-5-1-basic-trigonometry-edexcel-a-level-pure-maths-medium

Find the length AC, to the nearest m.

9b
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3 marks

Find the angle between the fences AD and CD.

9c
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3 marks

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

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10a
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3 marks

A triangle ABC has sides A B equals x space c m,A C equals space open parentheses x plus 8 close parentheses cm and angle BAC =150 to the power of degree

The area of the triangle is 2 1 fourthcm2.

Show that x satisfies the equation x squared plus 8 x equals 9

10b
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2 marks

Hence, or otherwise find the value of x

10c
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3 marks

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

10d
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3 marks

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

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11a
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2 marks

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.

q-11a-5-1-basic-trigonometry-edexcel-a-level-pure-maths-medium

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

11b
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3 marks

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.

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12a
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2 marks

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.

12b
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2 marks

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

12c
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2 marks

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

12d
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2 marks

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.

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1a
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2 marks

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

1b
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2 marks

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.

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2
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4 marks

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

q2-5-1-basic-trigonometry-edexcel-a-level-pure-maths-hard

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3a
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3 marks

A triangle ABC has side lengths AB = 2 x cm, BC = open parentheses x plus 5 close parentheses cm and AC = 7 cm. The angle ABC = 60°

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

3b
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3 marks

The area of the triangle can be written in the form a square root of 6 plus b square root of 3, where a and b are integers. Find the values of a and b

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4
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4 marks

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

(i)

     Find the value of cos space theta for the each of the angles in the triangle.

(ii)
Between which two sides is the biggest angle?

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5
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4 marks

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.

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6
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4 marks

 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?

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7
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4 marks

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.

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

(ii)
Given that Mast A and Mast B are the furthest distance apart of all three masts, find the distance between them.

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8a
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4 marks

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

Find the two possible angles for theta .

8b
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3 marks

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

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9a
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5 marks

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

q9a-5-1-basic-trigonometry-edexcel-a-level-pure-maths-hard

Find the angle between the fences BC and CD.

9b
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4 marks

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

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10a
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3 marks

A triangle ABC has sides A B equals 2 x space c m, A C equals open parentheses x plus 6 close parentheses space c m and angle B A C equals 30 to the power of degree.

The area of the triangle is  5 begin inline style 3 over 4 end style cm2.

Show that x satisfies the equation 2 x squared plus 12 x minus 23 equals 0.

10b
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3 marks

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

10c
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3 marks

Given angle ABC is obtuse, find the ratio of the angles of the triangle, to the nearest degree.

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11a
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2 marks

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

q11a-5-1-basic-trigonometry-edexcel-a-level-pure-maths-hard

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.

11b
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4 marks

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 x to the power of degree to 1 decimal place

11c
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3 marks

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.

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1
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4 marks

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

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2a
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2 marks

A triangle ABC has side lengths AB = 3 x cm, BC = 5 x cm and AC = 6x cm.

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

2b
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4 marks

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

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3a
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2 marks

In a triangle ABC, A B equals 2 x cm, B C equals 10 cm and A C equals open parentheses 20 minus 2 x close parentheses cm, angle A B C equals theta°.

Show that cos space theta equals fraction numerator 4 x minus 15 over denominator 2 x end fraction .

3b
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4 marks

Given that cos space theta equals negative 1 half , find the area of the triangle.

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4a
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3 marks

Triangle PSQ and SQR are such that PS = SQ = QR. Sides PQ = 15 cm and QR = 2 x cm. Angle PSQ = 120°.

q4a-5-1-basic-trigonometry-edexcel-a-level-pure-maths-veryhard

 Calculate the exact value of x.

4b
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2 marks

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

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5
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5 marks

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.

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6
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9 marks

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.

q6-5-1-basic-trigonometry-edexcel-a-level-pure-maths-veryhard

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

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7a
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4 marks

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.

7b
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1 mark

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

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8
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5 marks

A triangle ABC has sides AB = x cm, BC = open parentheses 4 minus x close parentheses cm, angle BAC = theta and angle BCA = 30°.

Given that sin space theta equals fraction numerator 1 over denominator square root of 2 end fraction , show that x equals 4 open parentheses square root of 2 minus 1 close parentheses .

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9
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6 marks

A triangle ABC has sides A B equals 3 x space c m,A C equals open parentheses x plus 5 close parentheses c m  and angle BACequals 150 degree .

The area of the triangle is 7 begin inline style 1 fourth end stylecm2.

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

Leave your answer in simplest form.

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