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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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Cards in this collection (11)

  • 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 open parentheses theta over 2 close parentheses as cos open parentheses theta space minus space theta over 2 close parentheses doesn’t change the ratio but could make an identity easier to prove.

  • To prove cos invisible function application 4 theta equals 8 cos to the power of 4 invisible function application theta minus 8 cos squared invisible function application theta plus 1, what identity should you substitute into the left-hand side?

    To prove cos invisible function application 4 theta equals 8 cos to the power of 4 invisible function application theta minus 8 cos squared invisible function application theta plus 1, the identity that you should substitute into the left-hand side is cos 2 theta equals 2 cos squared theta minus 1.

    The left-hand side implies the need to use the double angle formulae and the right-hand side involves cos squared theta, so suggests the use of 2 cos squared theta minus 1.

    (Note that you will need to substitute the version cos 4 theta equals 2 cos squared 2 theta minus 1 first, and then use cos 2 theta equals 2 cos squared theta minus 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 theta equals fraction numerator 1 over denominator tan theta end fraction, but you should be able to work this out from the other given identities for sec theta and cosec theta.

  • When solving a trigonometric equation, what should you do if the equation involves different multiples of x or theta?

    E.g. 3 cos open parentheses 2 x close parentheses minus cos space x equals 0

    When solving a trigonometric equation that involves different multiples of x or theta, you should use the double angle formulae or compound angle formulae to get everything in terms of the same multiple of x or theta.

  • When solving a trig equation that involves a function of x or theta, e.g. sin open parentheses 3 x minus 2 close parentheses equals 0, what should you do to the range?

    When solving a trig equation that involves a function of x or theta, e.g. sin open parentheses 3 x minus 2 close parentheses equals 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 theta equals fraction numerator sin theta over denominator cos theta end fraction.

  • True or False?

    If you have solved a trig equation for one of the reciprocal trig functions, e.g.cosec space x, you should convert it to a simple trig function, e.g. sin space x, before solving for x or theta.

    True.

    If you have solved a trig equation for one of the reciprocal trig functions, e.g.cosec space x, you should convert it to a simple trig function, e.g. sin space x, before solving for x or theta.

  • 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.

Previous:Inverse & Reciprocal Trigonometric FunctionsNext:Vector Properties

    1. Topic 1

      Number & Algebra Toolkit

    2. Topic 2

      Exponentials & Logs

    3. Topic 3

      Sequences & Series

    4. Topic 4

      Simple Proof & Reasoning

    5. Topic 5

      Proof by Induction & Contradiction

    6. Topic 6

      Binomial Theorem

    7. Topic 7

      Permutations & Combinations

    8. Topic 8

      Complex Numbers

    9. Topic 9

      Further Complex Numbers

    10. Topic 10

      Systems of Linear Equations

    1. Topic 1

      Linear Functions & Graphs

    2. Topic 2

      Quadratic Functions & Graphs

    3. Topic 3

      Functions Toolkit

    4. Topic 4

      Other Functions & Graphs

    5. Topic 5

      Reciprocal & Rational Functions

    6. Topic 6

      Transformations of Graphs

    7. Topic 7

      Polynomial Functions

    8. Topic 8

      Inequalities

    9. Topic 9

      Modulus Functions & Further Transformations

    1. Topic 1

      Geometry Toolkit

    2. Topic 2

      Geometry of 3D Shapes

    3. Topic 3

      Trigonometry

    4. Topic 4

      The Unit Circle & Exact Values

    5. Topic 5

      Trigonometric Functions & Graphs

    6. Topic 6

      Trigonometric Equations & Identities

    7. Topic 7

      Inverse & Reciprocal Trigonometric Functions

    8. Topic 8

      Trigonometric Proof & Equation Strategies

    9. Topic 9

      Vector Properties

    10. Topic 10

      Vector Equations of Lines

    11. Topic 11

      Vector Planes

    1. Topic 1

      Statistics Toolkit

    2. Topic 2

      Correlation & Regression

    3. Topic 3

      Probability

    4. Topic 4

      Discrete Random Variables

    5. Topic 5

      Binomial Distribution

    6. Topic 6

      Normal Distribution

    7. Topic 7

      Continuous Random Variables

    1. Topic 1

      Differentiation

    2. Topic 2

      Techniques & Applications of Differentiation

    3. Topic 3

      Integration

    4. Topic 4

      Techniques & Applications of Integration

    5. Topic 5

      Optimisation

    6. Topic 6

      Kinematics

    7. Topic 7

      Basic Limits & Continuity

    8. Topic 8

      Further Differentiation

    9. Topic 9

      Further Integration

    10. Topic 10

      Differential Equations

    11. Topic 11

      Maclaurin Series

    12. Topic 12

      Limits using l'Hôpital's Rule & Maclaurin Series