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Further Trigonometric Identitiesand their
ApplicationsIntroductionThis chapter extends your knowledge of
Trigonometrical identities
You will see how to solve equations involving combinations of
sin, cos and tan
You will learn to express combinations of these as a
transformation of a single graphTeachings for Exercise 7AFurther
Trigonometric Identities and their ApplicationsYou need to know and
be able to use the addition formulae7AQPN11ABBy GCSE
Trigonometry:OSo the coordinates of P are:MSo the coordinates of Q
are:QP4Further Trigonometric Identities and their ApplicationsYou
need to know and be able to use the addition formulae7AMultiply out
the bracketsRearrange5Further Trigonometric Identities and their
ApplicationsYou need to know and be able to use the addition
formulae7AQPN11ABOMYou can also work out PQ using the triangle
OPQ:B - A11QPSub in the valuesGroup termsCos (B A) = Cos (A B) eg)
Cos(60) = Cos(-60)6Further Trigonometric Identities and their
ApplicationsYou need to know and be able to use the addition
formulae7ASubtract 2 from both sidesDivide by -2Cos(A - B) =
CosACosB + SinASinBCos(A + B) = CosACosB - SinASinB7Further
Trigonometric Identities and their ApplicationsYou need to know and
be able to use the addition formulae7ACos(A - B) CosACosB +
SinASinBCos(A + B) CosACosB - SinASinBSin(A + B) SinACosB +
CosASinBSin(A - B) SinACosB - CosASinB8Further Trigonometric
Identities and their ApplicationsYou need to know and be able to
use the addition formulae
Show that:7ACos(A - B) = CosACosB + SinASinBCos(A + B) =
CosACosB - SinASinBSin(A + B) = SinACosB + CosASinBSin(A - B) =
SinACosB - CosASinBTanA+ TanB1- TanATanBRewriteDivide top and
bottom by CosACosBSimplify each Fraction9Further Trigonometric
Identities and their ApplicationsYou need to know and be able to
use the addition formulae7ACos(A - B) CosACosB + SinASinBCos(A + B)
CosACosB - SinASinBSin(A + B) SinACosB + CosASinBSin(A - B)
SinACosB - CosASinBYou may be asked to prove either of the Tan
identities using the Sin and Cos ones!10Further Trigonometric
Identities and their ApplicationsYou need to know and be able to
use the addition formulae
Show, using the formula for Sin(A B), that:7ACos(A - B) CosACosB
+ SinASinBCos(A + B) CosACosB - SinASinBSin(A + B) SinACosB +
CosASinBSin(A - B) SinACosB - CosASinBSin(A - B) SinACosB -
CosASinBSin(45 - 30) Sin45Cos30 Cos45Sin30Sin(45 - 30) Sin(45 - 30)
Sin(15) A=45, B=30These can be written as surdsMultiply each
pairGroup the fractions up11Further Trigonometric Identities and
their ApplicationsYou need to know and be able to use the addition
formulae
Given that:
Find the value of:7ATan(A+B)AB35412135Use Pythagoras to find the
missing side (ignore negatives)Tan is positive in the range 180 -
270Use Pythagoras to find the missing side (ignore negatives)Tan is
negative in the range 90 - 18090180270360y = Tan12Further
Trigonometric Identities and their ApplicationsYou need to know and
be able to use the addition formulae
Given that:
Find the value of:7ATan(A+B)Substitute in TanA and TanBWork out
the Numerator and DenominatorLeave, Change and FlipSimplifyAlthough
you could just type the whole thing into your calculator, you still
need to show the stages for the workings marks13Further
Trigonometric Identities and their ApplicationsYou need to know and
be able to use the addition formulaeGiven that:
Express Tanx in terms of Tany7ARewrite the sin and cos
partsMultiply out the bracketsDivide all by
cosxcosySimplifySubtract 3tanxtanySubtract 2tanyFactorise the left
sideDivide by (2 3tany)14Teachings for Exercise 7BFurther
Trigonometric Identities and their ApplicationsYou can express
sin2A, cos 2A and tan2A in terms of angle A, using the double angle
formulae7BSin(A + B) SinACosB + CosASinBSin(A + A) SinACosA +
CosASinASin2A 2SinACosAReplace B with ASimplifySin2A 2SinACosASin4A
2Sin2ACos2ASin60 2Sin30Cos303Sin2A 6SinACosA 22A 4Ax 32A =
60Further Trigonometric Identities and their ApplicationsYou can
express sin2A, cos 2A and tan2A in terms of angle A, using the
double angle formulae7BCos(A + B) CosACosB - SinASinBCos(A + A)
CosACosA - SinASinAReplace B with ASimplifyReplace Sin2A with (1
Cos2A)Replace Cos2A with (1 Sin2A)Further Trigonometric Identities
and their ApplicationsYou can express sin2A, cos 2A and tan2A in
terms of angle A, using the double angle formulae7BReplace B with
ASimplify 22A = 60x 22A = AFurther Trigonometric Identities and
their ApplicationsYou can express sin2A, cos 2A and tan2A in terms
of angle A, using the double angle formulae
Rewrite the following as a single Trigonometric function:7B2
Replace the first partRewriteFurther Trigonometric Identities and
their ApplicationsYou can express sin2A, cos 2A and tan2A in terms
of angle A, using the double angle formulae
Show that:
Can be written as:7BDouble the angle partsReplace cos4The 1s
cancel outFurther Trigonometric Identities and their
ApplicationsYou can express sin2A, cos 2A and tan2A in terms of
angle A, using the double angle formulae
Given that:
Find the exact value of:7BxUse Pythagoras to find the missing
side (ignore negatives)Cosx is positive so in the range 270 -
36034Therefore, Sinx is negative90180270360y = Siny = CosSin2x
2SinxCosxSub in Sinx and CosxWork out and leave in surd formFurther
Trigonometric Identities and their ApplicationsYou can express
sin2A, cos 2A and tan2A in terms of angle A, using the double angle
formulae
Given that:
Find the exact value of:7BxUse Pythagoras to find the missing
side (ignore negatives)Cosx is positive so in the range 270 -
36034Therefore, Tanx is negative90180270360y = CosSub in TanxWork
out and leave in surd form90180270360y = TanTeachings for Exercise
7CFurther Trigonometric Identities and their ApplicationsThe double
angle formulae allow you to solve more equations and prove more
identities
Prove the identity:7CDivide each part by tanRewrite each
partFurther Trigonometric Identities and their ApplicationsThe
double angle formulae allow you to solve more equations and prove
more identities
By expanding:
Show that:7CReplace A and BReplace Sin2A and Cos 2AMultiply
outReplace cos2AMultiply outGroup like termsFurther Trigonometric
Identities and their ApplicationsThe double angle formulae allow
you to solve more equations and prove more identities
Given that:
Eliminate and express y in terms of x7CandDivide by 3Subtract 3,
divide by 4Multiply by -1Replace Cos2 and SinMultiply by 4Subtract
3Multiply by -1Further Trigonometric Identities and their
ApplicationsThe double angle formulae allow you to solve more
equations and prove more identities
Solve the following equation in the range stated:
(All trigonometrical parts must be in terms x, rather than
2x)7CReplace cos2xMultiply out the bracketGroup
termsorFactorise90180270360y = CosSolve both pairsRemember to find
additional answers!Teachings for Exercise 7DFurther Trigonometric
Identities and their ApplicationsYou can write expressions of the
form acos + bsin, where a and b are constants, as a sine or cosine
function only
Show that:
Can be expressed in the form:
So:7DSo in the triangle, the Hypotenuse is RReplace with the
expressionCompare each term they must be equal!R = 5Inverse CosFind
the smallest value in the acceptable range givenFurther
Trigonometric Identities and their ApplicationsYou can write
expressions of the form acos + bsin, where a and b are constants,
as a sine or cosine function only
Show that you can express:
In the form:
So:
7DR = 2Divide by 2Inverse cosFind the smallest value in the
acceptable rangeReplace with the expressionCompare each term they
must be equal!Further Trigonometric Identities and their
ApplicationsYou can write expressions of the form acos + bsin,
where a and b are constants, as a sine or cosine function only
Show that you can express:
In the form:
So:
7DSketch the graph of:= Sketch the graph
of:/23/221-1/23/221/23/221-1/34/3-12-2Start out with sinxTranslate
/3 units rightVertical stretch, scale factor 2/34/3At the
y-intercept, x = 0Further Trigonometric Identities and their
ApplicationsYou can write expressions of the form acos + bsin,
where a and b are constants, as a sine or cosine function only
Express:
in the form:
So:
7DReplace with the expressionCompare each term they must be
equal!R = 29Divide by 29Inverse cosFind the smallest value in the
acceptable rangeFurther Trigonometric Identities and their
ApplicationsYou can write expressions of the form acos + bsin,
where a and b are constants, as a sine or cosine function only
Solve in the given range, the following equation:
7DWe just showed that the original equation can be
rewrittenHence, we can solve this equation instead!Remember to
adjust the range for ( 68.2)Divide by 29Inverse CosRemember to work
out other values in the adjusted rangeAdd 68.2 (and put in
order!)90180270360y = Cos-9056.1-56.1303.9Further Trigonometric
Identities and their ApplicationsYou can write expressions of the
form acos + bsin, where a and b are constants, as a sine or cosine
function only
Find the maximum value of the following expression, and the
smallest positive value of at which it arises:
7DReplace with the expressionCompare each term they must be
equal!R = 13Divide by 13Inverse cosFind the smallest value in the
acceptable rangeMax value of cos( - 22.6) = 1Overall maximum
therefore = 13Cos peaks at 0 = 22.6 gives us 0Rcos( ) chosen as it
gives us the same form as the expressionFurther Trigonometric
Identities and their ApplicationsYou can write expressions of the
form acos + bsin, where a and b are constants, as a sine or cosine
function only7DWhichever ratio is at the start, change the
expression into a function of that (This makes solving problems
easier)Remember to get the + or signs the correct way
round!Teachings for Exercise 7EFurther Trigonometric Identities and
their ApplicationsYou can express sums and differences of sines and
cosines as products of sines and cosines by using the factor
formulae7EYou get given all these in the formula booklet!Further
Trigonometric Identities and their ApplicationsYou can express sums
and differences of sines and cosines as products of sines and
cosines by using the factor formulae7EUsing the formulae for Sin(A
+ B) and Sin (A B), derive the result that:Add both sides together
(1 + 2)1)2)Let (A+B) = P Let (A-B) = Q1)2)1)2)1 + 2Divide by 21 -
2Divide by 2Further Trigonometric Identities and their
ApplicationsYou can express sums and differences of sines and
cosines as products of sines and cosines by using the factor
formulae7EShow that:P = 105 Q = 15Work out the fraction partsSub in
values for Cos60 and Sin45Work out the right hand sideFurther
Trigonometric Identities and their ApplicationsYou can express sums
and differences of sines and cosines as products of sines and
cosines by using the factor formulae7ESolve in the range
indicated:P = 4 Q = 3Work out the fractionsSet equal to 0Either the
cos or sin part must equal 0Inverse cosSolve, remembering to take
into account the different rangeOnce you have all the values from
0-2, add 2 to them to obtain equivalentsMultiply by 2 and divide by
7Adjust the range/23/22y = Cos0Further Trigonometric Identities and
their ApplicationsYou can express sums and differences of sines and
cosines as products of sines and cosines by using the factor
formulae7ESolve in the range indicated:P = 4 Q = 3Work out the
fractionsSet equal to 0Either the cos or sin part must equal
0Inverse sinSolve, remembering to take into account the different
rangeOnce you have all the values from 0-2, add 2 to them to obtain
equivalentsMultiply by 2Adjust the range/23/22y = Sin0Further
Trigonometric Identities and their ApplicationsYou can express sums
and differences of sines and cosines as products of sines and
cosines by using the factor formulae
Prove that:7EIn the numerator:Ignore sin(x + y) for nowUse the
identity for adding 2 sinesP = x + 2y Q = xSimplify FractionsBring
back the sin(x + y) we ignored earlierFactoriseNumerator:Further
Trigonometric Identities and their ApplicationsYou can express sums
and differences of sines and cosines as products of sines and
cosines by using the factor formulae
Prove that:7EIn the denominator:Ignore cos(x + y) for nowUse the
identity for adding 2 cosinesP = x + 2y Q = xSimplify
FractionsBring back the cos(x + y) we ignored
earlierFactoriseNumerator:Denominator:Further Trigonometric
Identities and their ApplicationsYou can express sums and
differences of sines and cosines as products of sines and cosines
by using the factor formulae
Prove that:7ENumerator:Denominator:Replace the numerator and
denominatorCancel out the (2cosy + 1) bracketsUse one of the
identities from C2SummaryWe have extended the range of techniques
we have for solving trigonometrical equations
We have seen how to combine functions involving sine and cosine
into a single transformation of sine or cosine
We have learnt several new identities
