IGCSEAdditional MathsCIETopic QuestionsTrigonometryTrigonometry

Trigonometry (CIE IGCSE Additional Maths)

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1
Sme Calculator3 marks

Solve the equation cot open parentheses y minus pi over 2 close parentheses space equals space square root of 3, where y is in radians and 0 space less-than or slanted equal to space y space less-than or slanted equal to space pi .

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2
Sme Calculator5 marks

Solve the equation 1 half sec open parentheses 2 ϕ plus pi over 4 close parentheses space equals space fraction numerator 1 over denominator square root of 3 end fraction for negative pi less than ϕ less than straight pi, where ϕ is in radians. Give your answers in terms of pi.

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3a
Sme Calculator2 marks

In this question all lengths are in centimetres.

q5a-0606-m20-qp-12-additional-maths

The diagram shows the isosceles triangle A B C, where A B space equals space A C and B C space equals space 2 plus 4 space square root of 3. The height, A D, of the triangle is 5 minus square root of space 3 end root.

Find tan A B C, giving your answer in the formspace c plus d square root of 3, where c space and space d are integers.

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3b
Sme Calculator2 marks

Find sec squared space A B C, giving your answer in the form e plus f square root of 3, where e space and space f are integers.

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4
Sme Calculator4 marks

Solve cosec space open parentheses y plus straight pi over 3 close parentheses equals 2 space for space 0 less or equal than y less or equal than 2 straight pi radians, giving your answers in terms of straight pi.

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5a
Sme Calculator4 marks

Solve tan space 3 x space equals negative 1 space for space minus straight pi over 2 less or equal than x less or equal than straight pi over 2 radians, giving your answers in terms of straight pi.

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5b
Sme Calculator3 marks

Use your answers to part (a) to sketch the graph of y space equals space 4 space tan space 3 x plus 4 space for space minus straight pi over 2 less or equal than x less or equal than straight pi over 2 radians on the axes below. Show the coordinates of the points where the curve meets the axes.

q10b-0606-s20-qp-12-additional-maths

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6
Sme Calculator5 marks

Solve 3 space cot squared space x minus 14 space cosec space x minus 2 space equals space 0 space for space 0 degree space less than space x space less than space 360 degree.

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7
Sme Calculator4 marks

Solve the equation

5 space sec squared space A plus 14 tan space A minus 8 space equals space 0 space for space 0 degree space less or equal than space A space less or equal than space 180 degree,

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8
Sme Calculator4 marks

Solve sin open parentheses 3 ϕ plus fraction numerator 2 straight pi over denominator 3 end fraction close parentheses equals cos open parentheses 3 ϕ plus fraction numerator 2 straight pi over denominator 3 end fraction close parentheses space for space 0 less or equal than ϕ less or equal than fraction numerator 2 straight pi over denominator 3 end fraction space radians, giving your answers in terms of straight pi.

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1a
Sme Calculator1 mark

In this question, all angles are in radians.

Write down the amplitude of 2 space cos space x over 3 minus 1.

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1b
Sme Calculator2 marks

Write down the period of 2 space cos space x over 3 minus 1.

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1c
Sme Calculator3 marks

On the axes below, sketch the graph of y equals 2 space cos space x over 3 minus 1 for – pi space less-than or slanted equal to space x space less-than or slanted equal to space 3 pi .

q3c-2025-specimen-paper-1-cie-igcse-additional-maths

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2
Sme Calculator5 marks
(i)
Write 6 x y plus 3 y plus 4 x plus 2 in the form left parenthesis a x plus b right parenthesis left parenthesis c y plus d right parenthesis, where a comma space b comma space c and d are positive integers.

(ii)
Hence solve the equation 6 space sin space theta space cos space theta plus 3 space cos space theta plus 4 space sin space theta plus 2 space equals space 0 for 0 degree space less than space theta space less than space 360 degree.

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3a
Sme Calculator3 marks

The graph of y space equals space a plus 2 space tan space b x, where a and b are constants, passes through the point left parenthesis 0 comma space minus 4 right parenthesis and has period 480 degree.

Find the value of a and of b.

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3b
Sme Calculator2 marks

On the axes, sketch the graph of y for values of x between 0° and 480°.

q4b-june-2021-paper-21-cie-igcse-additional-maths

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4
Sme Calculator3 marks

Solve tan open parentheses alpha plus 45 to the power of degree close parentheses equals negative fraction numerator 1 over denominator square root of 2 end fraction space for space 0 to the power of degree less or equal than alpha less or equal than 360 to the power of degree.

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5
Sme Calculator4 marks

q8-0606-m20-qp-22-additional-maths

The diagram shows the graph of straight f open parentheses x close parentheses equals a space cos space b x space plus c space for space 0 less or equal than x less or equal than fraction numerator 8 pi over denominator 3 end fractionradians.

Find the value of each of the constants a comma space b and c.

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6
Sme Calculator3 marks

The curve y space equals space a space sin space b x plus c has a period of 180°, an amplitude of 20 and passes through the point left parenthesis 90 degree comma space minus 3 right parenthesis. Find the value of each of the constants a comma space b space and space c.

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7
Sme Calculator2 marks

The function straight g is defined, forspace minus 135 degree space less or equal than space x space less or equal than space 135 degree comma space by space straight g left parenthesis x right parenthesis space equals space 3 space tan space x over 2 space minus 4. Sketch the graph of y space equals space straight g left parenthesis x right parenthesis on the axes below, stating the coordinates of the point where the graph crosses the y-axis.q6b-0606-s20-qp-23-additional-maths

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8
Sme Calculator4 marks

Solve the equation
5 space sin open parentheses 4 B minus straight pi over 8 close parentheses plus 2 equals 0 space for space minus straight pi over 4 less or equal than B less or equal than straight pi over 4 space radians

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9a
Sme Calculator1 mark

Write down the amplitude of 1 plus 4 space cos open parentheses x over 3 close parentheses

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9b
Sme Calculator1 mark

Write down the period of 1 plus 4 space cos open parentheses x over 3 close parentheses

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9c
Sme Calculator3 marks

On the axes below, sketch the graph of y equals 1 plus 4 space cos open parentheses x over 3 close parentheses space for space minus 180 degree less or equal than x degree less or equal than 180 degree.

q1c-0606-w20-qp-11-additional-maths

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1
Sme Calculator6 marks
(i)
Show that fraction numerator cos squared 2 x over denominator 1 plus space sin space 2 x end fraction equals 1 minus sin space 2 x.

(ii)
Hence solve the equation for fraction numerator 3 space cos squared space 2 x over denominator 1 plus sin space 2 x end fraction equals 1 for 0 degree space less-than or slanted equal to space x space less-than or slanted equal to space 90 degree .

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2
Sme Calculator8 marks
(i)
Show that fraction numerator 1 over denominator sin space theta space minus 1 end fraction minus fraction numerator 1 over denominator sin space theta space plus 1 end fraction equals a space sec squared theta, where a is a constant to be found.
[3]

(ii)
Hence solve fraction numerator 1 over denominator sin 3 ϕ minus 1 end fraction minus fraction numerator 1 over denominator sin 3 ϕ plus 1 end fraction equals negative 8 space for space minus pi over 3 less or equal than ϕ less or equal than pi over 3 space radians.
[5]

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3
Sme Calculator8 marks
(i)
Show that fraction numerator 1 over denominator sec theta minus 1 end fraction minus fraction numerator 1 over denominator sec theta plus 1 end fraction equals 2 cot squared theta.
[3]

(ii)
Hence solve fraction numerator 1 over denominator sec 2 x minus 1 end fraction minus fraction numerator 1 over denominator sec 2 x plus 1 end fraction equals 6 for negative 90 degree less than x less than 90 degree.
[5]

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4
Sme Calculator4 marks

Show that fraction numerator sin to the power of 4 y minus cos to the power of 4 y over denominator cot space y end fraction equals tan space y minus 2 space cos space y space sin space y.

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5
Sme Calculator8 marks

(i)
Show that fraction numerator 1 over denominator open parentheses 1 plus cosec space theta close parentheses open parentheses sin space theta minus sin to the power of 2 space end exponent theta close parentheses end fraction equals sec squared space theta space.

[4]

(ii)
Hence solve open parentheses 1 plus cosec space theta close parentheses open parentheses sin space theta minus sin squared theta close parentheses equals 3 over 4 space for space minus 180 degree less or equal than theta less or equal than 180 degree.

[4]

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6
Sme Calculator4 marks

Show that fraction numerator sin space x space tan space x over denominator 1 minus cos space x end fraction equals 1 plus sec space x.

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7a
Sme Calculator3 marks

Given that 2 space cos space x space equals space 3 space tan space x, show that 2 space sin squared x plus 3 space sin space x minus 2 space equals space 0.

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7b
Sme Calculator4 marks

Hence solve 2 space cos open parentheses 2 alpha plus straight pi over 4 close parentheses equals 3 space tan open parentheses 2 alpha plus straight pi over 4 close parentheses space for space 0 less than alpha less than straight pi radians, giving your answers in terms of straight pi.

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8
Sme Calculator6 marks

Solve the equation 5 space tan space x minus 3 space cot space x space equals space 2 space sec space x space for space 0 degree space less or equal than space x space less or equal than space 360 degree.

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