DP IB Maths: AA SL

Topic Questions

3.6 Trigonometric Equations & Identities

1
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6 marks

The value of sin space alpha equals 3 over 7  for  0 less or equal than alpha less or equal than pi over 2.  Find:

(i)
cos space alpha
(ii)
sin space 2 alpha
(iii)
cos space 2 alpha
(iv)
tan space 2 alpha.

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2
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6 marks

The value of cos space B equals space 1 fifth,  for  fraction numerator 3 straight pi over denominator 2 end fraction less or equal than B less or equal than 2 pi.  Find:

(i)
cos space 2 B
(ii)
sin space 2 B
(iii)
tan space 2 B.

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3
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4 marks

An angle M has the properties such that sin space M space equals space r and sin space 2 M space equals space s.  Find, in terms of r and s comma an expression for:

(i)
cos space M
(ii)
tan space M.

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4
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3 marks

Solve the equation 2 space sin space 2 theta equals 1 for  0 degree less or equal than theta less or equal than 360 degree.

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5
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5 marks

Solve the equation 2 space sin space x equals fraction numerator 1 over denominator sin space x end fraction for  0 degree less or equal than x less or equal than 360 degree.

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6a
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2 marks

Show that left parenthesis x plus 1 right parenthesis left parenthesis x minus 2 right parenthesis left parenthesis x minus 3 right parenthesis equals x cubed minus 4 x squared plus x plus 6.

6b
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5 marks

Use your result from part (a) to solve the equation

tan cubed space x minus 4 tan squared space x plus tan space x plus 6 equals 0

in the interval 0 degree less or equal than x less or equal than 360 degree.

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7a
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2 marks

Show that the equation 2 space sin squared space x plus 3 space cos space x equals 0 can be written in the form a space cos squared space x plus b space cos space x plus c equals 0, where a comma space b and c are integers to be found.

7b
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3 marks

Hence, or otherwise, solve the equation 2 space sin squared space x plus 3 space cos space x space equals space 0
for negative 180 degree less or equal than x less or equal than 180 degree.

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8a
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1 mark

Show that the equation

2 space cos squared space x minus sin space x equals 1

can be written in the form

2 space sin squared space x plus sin space x minus 1 equals 0

8b
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5 marks

Hence, solve the equation 2 space cos squared space x minus sin space x equals 1, for  0 less or equal than x less or equal than 4 pi

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9a
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4 marks

The graph below shows the function y equals f left parenthesis x right parenthesis where  f left parenthesis x right parenthesis equals cos space x space space for space minus pi less or equal than x less or equal than pi.

q3a-3-6-medium-ib-aa-sl-maths

The function g left parenthesis x right parenthesis is formed by translating the function f left parenthesis x right parenthesis 1 unit vertically downwards.

The function h left parenthesis x right parenthesis is formed by stretching the function f left parenthesis x right parenthesis by a factor of 1 half in the y direction. The domain of h left parenthesis x right parenthesis remains the same as f left parenthesis x right parenthesis.

(i)
Sketch the functions y equals h left parenthesis x right parenthesis and y equals g left parenthesis x right parenthesis.
(ii)
State the number of roots for g left parenthesis x right parenthesis.
9b
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4 marks

Find the solutions to the equation cos space 2 x equals cos space x minus 1, for space minus pi less or equal than x less or equal than pi comma  and label them clearly on the graph of y equals f left parenthesis x right parenthesis given above.

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