True or False? To prove an identity you must always work on the left-hand side and proceed step by step until you achieve the expression on the right-hand side.

False. To prove an identity you can work from either side . It is more common to start on the left-hand side, but you can start a proof from the right-hand side. You should not work on both sides simultaneously .

True or False? An effective strategy for trigonometric proof is to start by reducing the number of different trigonometric functions there are within a given identity.

True. An effective strategy for trigonometric proof is to start by reducing the number of different trigonometric functions there are within a given identity.

True or False? You should calculate with fractions rather than decimals in trigonometric proof .

True. Generally speaking, you should calculate with fractions rather than decimals in trigonometric proof as there will be less scope for error and you can more easily express answers with exact values. You need to be confident in working with fractions and fractions-within-fractions.

What two tools can you use to find all solutions in the specified range ?

The two tools can you use to find all solutions in the specified range are: the CAST diagram, and the trigonometric graphs.

IBMathsDPAnalysis & Approaches (AA)HLFlashcards3. Geometry & TrigonometryTrigonometric Proof & Equation Strategies

Trigonometric Proof & Equation Strategies (DP IB Analysis & Approaches (AA))

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FrontTrigonometric Proof
True or False?
To prove an identity you must always work on the left-hand side and proceed step by step until you achieve the expression on the right-hand side.

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True or False?
To prove an identity you must always work on the left-hand side and proceed step by step until you achieve the expression on the right-hand side.
False.
To prove an identity you can work from either side.
It is more common to start on the left-hand side, but you can start a proof from the right-hand side.
You should not work on both sides simultaneously.
True or False?
An effective strategy for trigonometric proof is to start by reducing the number of different trigonometric functions there are within a given identity.
True.
An effective strategy for trigonometric proof is to start by reducing the number of different trigonometric functions there are within a given identity.
True or False?
The compound angle formulae can be used to prove further trig identities.
True.
The compound angle formulae can be used to prove further trig identities.
For example, rewriting $\cos (\frac{θ}{2})$ as $\cos (θ - \frac{θ}{2})$ doesn’t change the ratio but could make an identity easier to prove.
To prove $\cos 4 θ = 8 \cos^{4} θ - 8 \cos^{2} θ + 1$ , what identity should you substitute into the left-hand side?
To prove $\cos 4 θ = 8 \cos^{4} θ - 8 \cos^{2} θ + 1$ , the identity that you should substitute into the left-hand side is $\cos 2 θ = 2 \cos^{2} θ - 1$ .
The left-hand side implies the need to use the double angle formulae and the right-hand side involves $\cos^{2} θ$ , so suggests the use of $2 \cos^{2} θ - 1$ .
(Note that you will need to substitute the version $\cos 4 θ = 2 \cos^{2} 2 θ - 1$ first, and then use $\cos 2 θ = 2 \cos^{2} θ - 1$ in a subsequent step.)
True or False?
You should calculate with fractions rather than decimals in trigonometric proof.
True.
Generally speaking, you should calculate with fractions rather than decimals in trigonometric proof as there will be less scope for error and you can more easily express answers with exact values.
You need to be confident in working with fractions and fractions-within-fractions.
True or False?
You need to remember the relevant trig formulae.
False.
You do not need to remember the relevant trig formulae as all relevant formulae are given in the exam formula booklet.
The only relevant trig identity not given to you is $\cot θ = \frac{1}{\tan θ}$ , but you should be able to work this out from the other given identities for $\sec θ$ and $cosec θ$ .
When solving a trigonometric equation, what should you do if the equation involves different multiples of $x$ or $θ$ ?
E.g. $3 \cos (2 x) - \cos x = 0$
When solving a trigonometric equation that involves different multiples of $x$ or $θ$ , you should use the double angle formulae or compound angle formulae to get everything in terms of the same multiple of $x$ or $θ$ .
When solving a trig equation that involves a function of $x$ or $θ$ , e.g. $\sin (3 x - 2) = 0$ , what should you do to the range?
When solving a trig equation that involves a function of $x$ or $θ$ , e.g. $\sin (3 x - 2) = 0$ , you should transform the range before solving.
You must also remember to transform your solutions back again at the end.
When solving a trigonometric equation, what should you do if the equation involves more than one trigonometric function?
If a trigonometric equation involves more than one trigonometric function, you should try to reduce the equation to a single simple identity, e.g. tan x.
You can do this by:
- re-arranging everything to one side and factorising (if required),
- or using a trig identity, e.g. $\tan θ = \frac{\sin θ}{\cos θ}$ .
True or False?
If you have solved a trig equation for one of the reciprocal trig functions, e.g. $cosec x$ , you should convert it to a simple trig function, e.g. $\sin x$ , before solving for $x$ or $θ$ .
True.
If you have solved a trig equation for one of the reciprocal trig functions, e.g. $cosec x$ , you should convert it to a simple trig function, e.g. $\sin x$ , before solving for $x$ or $θ$ .
What two tools can you use to find all solutions in the specified range?
The two tools can you use to find all solutions in the specified range are:
- the CAST diagram,
- and the trigonometric graphs.

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