Trigonometry

    AQA
    A-Level

    Trigonometry explores the mathematical relationships between the side lengths and angles of triangles, serving as a fundamental tool for geometric problem-solving and modeling periodic phenomena. The topic progresses from the application of trigonometric ratios (sine, cosine, and tangent) in right-angled triangles to the use of the Sine and Cosine rules for non-right-angled triangles. It further extends to calculating the area of triangles, analyzing trigonometric functions and their graphs, and solving complex problems in three-dimensional contexts.

    0
    Objectives
    3
    Exam Tips
    4
    Pitfalls
    7
    Key Terms
    5
    Mark Points

    What You Need to Demonstrate

    Key skills and knowledge for this topic

    • Award M1 for correct substitution of identities (e.g., replacing sin²x with 1-cos²x) to reduce an equation to a single trigonometric function
    • Award A1 for finding the correct principal value using inverse trigonometric functions
    • Award A1 for identifying all valid secondary solutions within the specified range, utilizing the symmetry or periodicity of the function
    • Award B1 for correctly stating small angle approximations: sin(θ) ≈ θ, cos(θ) ≈ 1 - θ²/2, tan(θ) ≈ θ
    • Award M1 for expressing a linear combination a*cos(x) + b*sin(x) in the harmonic form R*cos(x ± α)

    Example Examiner Feedback

    Real feedback patterns examiners use when marking

    • "You have correctly found the principal value, but you missed the second solution in the quadrant—sketch the graph to find it."
    • "Good use of the double angle formula. To ensure full marks, explicitly write down the formula before substituting values."
    • "You lost a solution by dividing by sin(x). Instead, factorise the equation to find where sin(x) = 0."
    • "Your proof is logically sound, but remember to conclude with 'LHS = RHS' to formally complete the argument."

    Marking Points

    Key points examiners look for in your answers

    • Award M1 for correct substitution of identities (e.g., replacing sin²x with 1-cos²x) to reduce an equation to a single trigonometric function
    • Award A1 for finding the correct principal value using inverse trigonometric functions
    • Award A1 for identifying all valid secondary solutions within the specified range, utilizing the symmetry or periodicity of the function
    • Award B1 for correctly stating small angle approximations: sin(θ) ≈ θ, cos(θ) ≈ 1 - θ²/2, tan(θ) ≈ θ
    • Award M1 for expressing a linear combination a*cos(x) + b*sin(x) in the harmonic form R*cos(x ± α)

    Examiner Tips

    Expert advice for maximising your marks

    • 💡When proving identities, always manipulate the more complex side (usually LHS) to match the simpler side; do not move terms across the equals sign as this invalidates the proof structure
    • 💡For 'Show that' questions, explicitly state the standard identity being used (e.g., 'Using sin²x + cos²x = 1') before substituting to secure method marks even if calculation errors follow
    • 💡Always sketch the relevant trigonometric graph when solving equations to visually verify the number of solutions expected in the interval

    Common Mistakes

    Pitfalls to avoid in your exam answers

    • Dividing an equation by a common trigonometric factor (e.g., dividing by cos(x)) without considering the case where cos(x) = 0, leading to lost solutions
    • Failure to adjust the domain when solving for a compound angle (e.g., solving sin(2x) = 0.5 for 0 < x < 360 requires checking the range 0 < 2x < 720)
    • Incorrect signs when expanding compound angles, particularly assuming cos(A + B) = cos(A) + cos(B) or forgetting the sign change in the expansion
    • Using degree mode on the calculator when evaluating small angle approximations or calculus-based trigonometric questions, which strictly require radians

    Key Terminology

    Essential terms to know

    Likely Command Words

    How questions on this topic are typically asked

    Solve
    Prove
    Show that
    Sketch
    Deduce
    Express

    Practical Links

    Related required practicals

    • •{"code":"Modelling","title":"Simple Harmonic Motion","relevance":"Using R*sin(x+a) to analyse oscillating systems"}
    • •{"code":"Modelling","title":"Tidal Patterns","relevance":"Predicting water depth using periodic functions"}

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