ks5 full coverage: trigonometric equations and expressions
TRANSCRIPT
www.drfrostmaths.com
KS5 "Full Coverage": Trigonometric Equations and
Expressions (Yr2)
This worksheet is designed to cover one question of each type seen in past papers, for each A
Level topic. This worksheet was automatically generated by the DrFrostMaths Homework
Platform: students can practice this set of questions interactively by going to
www.drfrostmaths.com, logging on, Practise β Past Papers (or Library β Past Papers for
teachers), and using the βRevisionβ tab.
Question 1 Categorisation: Double angle formula for πππ
[Edexcel C3 June 2009 Q8a] Write down π ππ 2π₯ in terms of π ππ π₯ and πππ π₯ .
π ππ 2π₯ = ..........................
Question 2 Categorisation: Simplify an expression involving a double angle formula.
[Edexcel C3 June 2016 Q8a Edited] Write 2 πππ‘ 2π₯ + π‘ππ π₯ , π₯ β ππ
2, π β β€
as a single trigonometric ratio.
2 πππ‘ 2π₯ + π‘ππ π₯ β‘ ..........................
www.drfrostmaths.com
Question 3 Categorisation: As above.
[Edexcel A2 Specimen Papers P2 Q13a Edited]
Show that
πππ ππ 2π₯ + πππ‘ 2π₯ β‘ πππ‘ π₯ , π₯ β 90πΒ°, π β β€
..........................
Question 4 Categorisation: As above.
[Edexcel C3 June 2017 Q9a Edited]
Prove that
π ππ 2π₯ β π‘ππ π₯ β‘ π‘ππ π₯ πππ 2π₯ π₯ β (2π + 1)90Β° , π β β€
..........................
www.drfrostmaths.com
Question 5 Categorisation: Recognise that the double angle formula for πππ ππ β‘ πππππ β πππππ
is the difference of two squares, allowing for possible further simplification.
[Edexcel C3 June 2015 Q8a] Prove that
π ππ 2π΄ + π‘ππ 2π΄ β‘ πππ π΄ + π ππ π΄
πππ π΄ β π ππ π΄
..........................
Question 6 Categorisation: Solve a trigonometric equation using that identities π + πππππ β‘
πππππ and π + πππππ β‘ πππππππ
[Edexcel C3 Jan 2012 Q5]
Solve, for 0 β€ π β€ 180Β° ,
2 πππ‘ 23π = 7 πππ ππ 3π β 5
Give your answers in degrees to 1 decimal place.
..........................
www.drfrostmaths.com
Question 7 Categorisation: Use the Addition Formulae an simplify an expression.
[Edexcel C3 June 2015 Q1c]
Given that π‘ππ πΒ° = π , where π is a constant, π β Β±1 , use standard trigonometric
identities, to find πππ‘ (π β 45)Β° in terms of π .
πππ‘ (π β 45)Β° = ..........................
Question 8 Categorisation: Use the Addition Formulae to find sin/cos/tan of 15, 75, etc.
[Edexcel C3 Jan 2009 Q6b Edited] Using π ππ (π β πΌ) = π ππ π πππ πΌ β πππ π π ππ πΌ , or
otherwise, show that
π ππ 15Β° =1
4(βπ₯ β βπ¦)
where π₯ and π¦ are integers to be found.
..........................
Question 9 Categorisation: Recognise that πππ(π) β‘ πππ(ππ β π)
[Edexcel C3 June 2013 Q3a Edited]
Given that 2 πππ (π₯ + 50)Β° = π ππ (π₯ + 40)Β° , show that π‘ππ π₯Β° = π π‘ππ 40Β° , where π is a
constant to be found.
..........................
www.drfrostmaths.com
Question 10 Categorisation: Use the double-angle formulae backwards.
[Edexcel C3 Jan 2013 Q6i]
Without using a calculator, find the exact value of
( π ππ 22.5Β° + πππ 22.5Β°)2
You must show each stage of your working.
..........................
Question 11 Categorisation: Use the Additional Formulae for πππ.
[Edexcel C3 Jan 2012 Q8b Edited] Given that
π‘ππ (π΄ + π΅) = π‘ππ π΄ + π‘ππ π΅
1 β π‘ππ π΄ π‘ππ π΅
Deduce that
π‘ππ (π +π
6) =
1 + π π‘ππ π
π + π π‘ππ π
where π , π and π are constants to be found.
..........................
www.drfrostmaths.com
Question 12 Categorisation: Solve an equation given a previously proven identity.
[Edexcel C3 June 2006 Q6c Edited]
It can be shown that ππ π 4π β πππ‘ 4π = ππ π 2π + πππ‘ 2π
Solve, for 90Β° < π < 180Β° ,
ππ π 4π β πππ‘ 4π = 2 β πππ‘ π
π = .......................... Β°
Question 13 Categorisation: As above.
[Edexcel C3 Jan 2012 Q8c Edited]
π‘ππ (π +π
6) =
1 + β3 π‘ππ π
β3 β π‘ππ π
Hence, or otherwise, solve, for 0 β€ π β€ π ,
1 + β3 π‘ππ π = (β3 β π‘ππ π) π‘ππ (π β π)
Give your answers as multiples of π .
..........................
www.drfrostmaths.com
Question 14 Categorisation: Triple angle formula for πππ and πππ.
[Edexcel C3 Jan 2009 Q6ai Edited]
By writing 3π = (2π + π) , show that
π ππ 3π = π π ππ π β π π ππ 3π
where π and π are constants to be found.
..........................
Question 15 Categorisation: As above.
[Edexcel C3 Jan 2008 Q6a]
Use the double angle formulae and the identity
πππ (π΄ + π΅) β‘ πππ π΄ πππ π΅ β π ππ π΄ π ππ π΅
to obtain an expression for πππ 3π₯ in terms of powers of πππ π₯ only.
πππ 3π₯ β‘ ..........................
www.drfrostmaths.com
Question 16 Categorisation: Use double angle formulae to convert parametric equations to a
Cartesian one.
[Edexcel C4 June 2013 Q4b Edited] A curve C has parametric equations
π₯ = 2 π ππ π‘ , π¦ = 1 β πππ 2π‘ , βπ
2β€ π‘ β€
π
2
Find a cartesian equation for C in the form
π¦ = π(π₯) , β2 β€ π₯ β€ 2
π¦ = ..........................
Question 17 Categorisation: Use small-angle approximations.
[Edexcel A2 Specimen Papers P2 Q2a Edited] Given that π is small, use the small angle
approximation for πππ π to show that
1 + 4 πππ π + 3 πππ 2π β π + ππ2
where π and π are constants to be found.
..........................
www.drfrostmaths.com
Question 18 Categorisation: As above
When π is small and measured in radians, find the approximate value of
π( π ππ π + π‘ππ π)
πππ π β 1
..........................
Question 19 Categorisation: As above.
[Edexcel A2 Specimen Papers P2 Q2bi Edited]
Given that π is small, using the small angle approximation for πππ π it can be shown that
1 + 4 πππ π + 3 πππ 2π β 8 β 5π2
Adele uses π = 5Β° to test this approximation.
Adeleβs working is shown below.
Identify the mistake made by Adele in her working.
..........................
www.drfrostmaths.com
Question 20 Categorisation: Write π πππ π + π πππ π in the form πΉ πππ(π + πΆ) or πΉ πππ (π + πΆ)
[Edexcel C3 June 2015 Q3a]
π(π) = 4 πππ 2π + 2 π ππ 2π
Given that (π) = π πππ (2π β πΌ) , where π > 0 and 0 < πΌ < 90Β° , find the exact value of π
and the value of πΌ to 2 decimal places.
..........................
Question 21 Categorisation: As above.
[Edexcel A2 Specimen Papers P1 Q13a]
Express 2 π ππ π β 1.5 πππ π in the form π π ππ (π β πΌ) , where π > 0 and 0 < πΌ <π
2
State the value of π and give the value of πΌ to 4 decimal places.
..........................
www.drfrostmaths.com
Question 22 Categorisation: As above, but used for modelling.
[Edexcel A2 Specimen Papers P1 Q13b Edited]
It can be shown that 2 π ππ π β 1.5 πππ π = 2.5 π ππ (π β 0.6435) where π is measured in
radians.
Tom models the depth of water, π· metres, at Southview harbour on 18th October 2017 by
the formula
π· = 6 + 2 π ππ (4ππ‘
25) β 1.5 πππ (
4ππ‘
25) , 0 β€ π‘ β€ 24
where π‘ is the time, in hours, after 00:00 hours on 18th October 2017.
Use Tomβs model to find the depth of water at 00:00 hours on 18th October 2017.
.......................... m
Question 23 Categorisation: Determine the maximum or minimum value of a trigonometric
expression.
[Edexcel A2 Specimen Papers P1 Q13c Edited] (Continued from above) Use Tomβs model to
find the maximum depth of water.
.......................... m
www.drfrostmaths.com
Question 24 Categorisation: Determine the time/value of π/π for which the maximum/minimum
occurs.
[Edexcel A2 Specimen Papers P1 Q13d Edited] (Continued from above)
Use Tomβs model to find the time, in the afternoon, when the maximum depth of
water occurs.
Give your answer to the nearest minute.
................ : ................
Question 25 Categorisation: Describe how a model would be adapted.
[Edexcel A2 SAM P2 Q13d Edited] It can be shown that 10 πππ π β 3 π ππ π =
β109 πππ (π + 16.70)
The height above the ground, π» metres, of a passenger on a Ferris wheel π‘ minutes after the
wheel starts turning, is modelled by the equation
π» = 11 β 10 πππ (80π‘)Β° + 3 π ππ (80π‘)Β°
Figure 3 shows the graph of π» against π‘ for two complete cycles of the wheel. It is decided
that, to increase profits, the speed of the wheel is to be increased. How would you adapt the
equation of the model to reflect this increase in speed?
www.drfrostmaths.com
Question 26 Categorisation: Determine the time taken for a trigonometric function to make one
or more βcyclesβ.
[Edexcel A2 SAM P2 Q13c Edited] (Continued from above) Find the time taken, to the
nearest second, for the passenger to reach the maximum height on the second cycle.
(Solutions based entirely on graphical or numerical methods are not acceptable.)
.......................... minutes
and .......................... seconds
Question 27 Categorisation: Determine the maximum or minimum value of an trigonometric
function used within another expression, e.g. a division.
[Edexcel C3 Jan 2013 Q4bi Edited]
6 πππ π + 8 π ππ π can be expressed as 10 πππ (π β 0.927) .
π(π) =4
12 + 6 πππ π + 8 π ππ π, 0 < π < 2π
Calculate the maximum value of π(π) .
Maximum = ..........................
www.drfrostmaths.com
Question 28 Categorisation: Consider the number of solutions of a trigonometric equation.
[Edexcel C3 June 2015 Q3c]
π(π) = 4 πππ 2π + 2 π ππ 2π
It can be shown that π(π) = β20 πππ (2π β 26.57Β°)
Given that π is a constant and the equation π(π) = π has no solutions, state the range of
possible values of π .
..........................
Question 29 Categorisation: Solve an equation involving arcsin or arccos.
[Edexcel C3 June 2016 Q7b]
π(π₯) = ππππ ππ π₯ , β1 β€ π₯ β€ 1
Find the exact value of π₯ for which
3π(π₯ + 1) + π = 0
..........................
www.drfrostmaths.com
Question 30 Categorisation: Put an expression π πππ π + π πππ π in the form πΉ πππ (π + πΆ) in a
context where it is not obvious that you might need to do so.
[Edexcel C4 June 2014(R) Q8c Edited]
The curve shown in Figure 3 has parametric equations
π₯ = π‘ β 4 π ππ π‘ , π¦ = 1 β 2 πππ π‘ , β2π
3β€ π‘ β€
2π
3
The point π΄ , with coordinates (π, 1) , lies on the curve.
There is one point on the curve where the gradient is equal to β1
2 .
Given that π = 4 βπ
2 , and that
ππ¦
ππ₯=
2 π ππ π‘
1β4 πππ π‘ , find the value of π‘ at this point, showing each
step in your working and giving your answer to 4 decimal places.
[Solutions based entirely on graphical or numerical methods are not acceptable.]
π‘ = ..........................
www.drfrostmaths.com
Answers
Question 1
π ππ 2π₯ = 2 π ππ π₯ πππ π₯
Question 2
2 πππ‘ 2π₯ + π‘ππ π₯ β‘ πππ‘ π₯
Question 3
πππ ππ 2π₯ + cot 2π₯
β‘1
sin 2π₯+
cos 2π₯
sin 2π₯
β‘1 + 2 cos2 π₯ β 1
2 sin π₯ cos π₯β‘
2 cos2 π₯
2 sin π₯ cos π₯
β‘cos π₯
sin π₯β‘ cot π₯
Question 4
sin 2π₯ β tan π₯ β‘ tan π₯ cos 2π₯
LHS = 2 sin π₯ cos π₯ βsin π₯
cos π₯
=2 sin π₯ cos2 π₯ β sin π₯
cos π₯
= (sin π₯
cos π₯) (2 cos2 π₯ β 1)
= tan π₯ cos 2π₯
Question 5
sec 2π΄ + tan 2π΄ β‘1
cos 2π΄+
sin 2π΄
cos 2π΄
β‘1 + 2 sin π΄ πππ π΄
cos2 π΄ β sin2 π΄
β‘sin2 π΄ + cos2 π΄ + 2 sin π΄ πππ π΄
(cos π΄ + sin π΄)(cos π΄ β sin π΄)
β‘(cos π΄ + sin π΄)2
(cos π΄ + sin π΄)(cos π΄ β sin π΄)
β‘cos π΄ + sin π΄
cos π΄ β sin π΄
(note that it is easier to work from both the LHS and RHS
and meet in the middle)
Question 6
π = 6.5 or π = 53.5 or π = 126.5 or π =
173.5
Question 7
πππ‘ (π β 45)Β° =1+π
πβ1
Question 8
π₯ = 6 , π¦ = 2
Question 9
π =1
3
Question 10
1 +β2
2
Question 11
π = β3 , π = β3 , π = β1
Question 12
π = 135 Β°
www.drfrostmaths.com
Question 13
π =5π
12 or π =
11π
12
Question 14
π = 3 , π = 4
Question 15
cos 3π₯
β‘ 4 πππ 3π₯ β 3 πππ π₯
Question 16
π¦ =π₯2
2
Question 17
π = 8 , π = β5
Question 18
β4
Question 19
"degree OR degrees OR radian OR radians"
Question 20
π = β20 , πΌ = 26.57
Question 21
π = 2.5 , πΌ = 0.6435
Question 22
4.5 m
Question 23
8.5 m
Question 24
4: 54
Question 25
Increase the 80 in the formula
Question 26
6 minutes and and 32 seconds
Question 27
Maximum = 2
Question 28
π > β20 or π < ββ20
Question 29
π₯ = β1 ββ3
2
Question 30
π‘ = 0.6077