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IB DP Maths: AA HL复习笔记3.8.2 Strategy for Trigonometric Equations
Category:
IB课程
,
教材笔记
,
福利干货
Date: 2022年7月12日 下午4:20
Strategy for Trigonometric Equations
How do I approach solving trig equations?
You can solve trig equations in a variety of different ways
Sketching a graph
If you have your G
DC
it is always worth sketching the graph and using this to analyse its features
Using
trigonometric
identities
, Pythagorean
identities
,
the
compound
or
double angle
identities
Almost all of these are in the formula booklet, make sure you have it open at the right page
Using the
unit circle
Factorising
quadratic
trig equations
Look out for quadratics such as 5tan
2
x
– 3tan
x
– 4 = 0
The final rearranged equation you solve will involve
sin
,
cos
or
tan
Don’t try to solve an equation with
cosec
,
sec
, or
cot
directly
What should I look for when solving trig equations?
Check the value of
x
or θ
If it is just
x
or θ you can begin solving
If there are
different multiples
of
x
or θ you will need to use the
double angle formulae
to get everything in terms of the same multiple of
x
or θ
If it is a
function
of
x
or θ, e.g. 2
x
– 15, you will need to
transform the range
first
You must remember to transform your solutions back again at the end
Does it involve more than one trigonometric function?
If it does, try to
rearrange
everything to bring it to one side, you may need to
factorise
If not, can you use an identity to reduce the number of different trigonometric functions?
You should be able to use identities to reduce everything to just one simple trig function (either sin, cos or tan)
Is it
linear
or
quadratic
?
If it is linear you should be able to rearrange and solve it
If it is quadratic you may need to factorise first
You will most likely get two solutions, consider whether they both
exist
Remember solutions to sin
x
=
k
and cos
x
=
k
only exist for -1 ≤
k
≤ 1 whereas solutions to tan
x
=
k
exist for all values of
k
Are my solutions within the given range and do I need to find more solutions?
Be extra careful if your solutions are negative but the given range is positive only
Use a sketch of the graph or the unit circle to find the other solutions within the range
If you have a function of
x
or θ make sure you are finding the solutions within the
transformed range
Don’t forget to transform the solutions back so that they are in the required range at the end
Exam Tip
Try to use identities and formulas to reduce the equation into its simplest terms.
Don’t forget to check the function range and ensure you have included all possible solutions.
If the question involves a function of x or θ ensure you transform the range first (and ensure you transform your solutions back again at the end!).
Worked Example
转载自savemyexams
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