Trigonometry

OCR

A-Level

Trigonometry quantifies the intrinsic relationships between side lengths and internal angles of triangles through the sine, cosine, and tangent ratios. Fundamental application involves right-angled triangles using SOH CAH TOA, extending to general triangles via the Sine and Cosine Rules and the area formula ½absinC. Advanced study includes the analysis of trigonometric functions as periodic graphs, transformations of these functions, and the resolution of complex spatial problems in three dimensions. Candidates must demonstrate precision in algebraic manipulation and the selection of appropriate axioms for geometric proof and calculation.

Objectives

Exam Tips

Pitfalls

Key Terms

Mark Points

Subtopics in this area

Trigonometry

What You Need to Demonstrate

Key skills and knowledge for this topic

M1: Award for correct substitution of Pythagorean identities (e.g., sin²x = 1 - cos²x) to form a solvable quadratic equation
A1: Award for obtaining all correct solutions within the specified range; deduct for extra incorrect solutions within range
B1: Award for correct statement and application of double angle formulae (e.g., cos 2x = 2cos²x - 1)
M1: Award for correct application of small angle approximations (sin θ ≈ θ, cos θ ≈ 1 - θ²/2) in limit questions
E1: Award for a fully logical proof where the Left Hand Side (LHS) is manipulated clearly to match the Right Hand Side (RHS) without moving terms across the equals sign
Award M1 for the correct substitution of identities (e.g., replacing 1 + tan²x with sec²x) to form a solvable equation
Award A1 for identifying all valid solutions within the specified range, including negative angles where applicable
Award B1 for correct asymptotic behaviour and intercepts when sketching reciprocal trigonometric graphs

Example Examiner Feedback

Real feedback patterns examiners use when marking

"You have correctly identified the identity, but you must show the algebraic simplification step to secure the 'Show that' mark"
"Be careful not to lose solutions—factorise the expression rather than dividing by the trigonometric term"
"Your value for α is correct, but ensure you give the answer to the specific accuracy requested (e.g., 3 significant figures)"
"Excellent use of the harmonic form; now apply this structure to find the maximum or minimum value of the function"
"You correctly identified the identity, but you lost solutions by dividing. Try factorizing next time."
"Your proof is logical, but you moved terms across the equals sign. Rewrite it working solely from the LHS."
"Excellent use of the harmonic form. To secure the final mark, ensure your value for α is to the specified degree of accuracy."
"You have found the principal solution correctly; now use the quadrant rule or graph to find the second solution in the range."
"You correctly identified the identity, but you lost a solution by dividing by sin x. Try factorising next time."
"Good use of the double angle formula. To secure the final mark, ensure you explicitly check all solutions against the adjusted interval."
"Your proof is logical, but you skipped the step showing the common denominator. In 'Show that' questions, this step is mandatory."
"Excellent application of the R-addition method. Remember to state the value of alpha to the specified degree of accuracy."

Marking Points

Key points examiners look for in your answers

M1: Award for correct substitution of Pythagorean identities (e.g., sin²x = 1 - cos²x) to form a solvable quadratic equation
A1: Award for obtaining all correct solutions within the specified range; deduct for extra incorrect solutions within range
B1: Award for correct statement and application of double angle formulae (e.g., cos 2x = 2cos²x - 1)
M1: Award for correct application of small angle approximations (sin θ ≈ θ, cos θ ≈ 1 - θ²/2) in limit questions
E1: Award for a fully logical proof where the Left Hand Side (LHS) is manipulated clearly to match the Right Hand Side (RHS) without moving terms across the equals sign
Award M1 for the correct substitution of identities (e.g., replacing 1 + tan²x with sec²x) to form a solvable equation
Award A1 for identifying all valid solutions within the specified range, including negative angles where applicable
Award B1 for correct asymptotic behaviour and intercepts when sketching reciprocal trigonometric graphs
Award M1 for the correct expansion of compound or double angle formulae (e.g., sin(A+B)) as a necessary step in proof or solution
Credit responses that clearly state the value of R and α in harmonic form questions, ensuring α is in the correct quadrant
Award M1 for correct substitution of identities (e.g., replacing sin²x with 1-cos²x) to form a solvable equation
Award A1 for obtaining the correct quadratic equation in a single trigonometric ratio
Award B1 for stating the correct principal value within the given range, strictly adhering to radian/degree constraints
Award M1 for the correct method in determining R and alpha, where R > 0 and alpha is acute
Credit responses that explicitly show the rejection of invalid solutions (e.g., cos x > 1) with a brief justification

Examiner Tips

Expert advice for maximising your marks

💡When the command phrase 'In this question you must show detailed reasoning' is used, you must explicitly write down the identities and intermediate algebraic steps; calculator-only answers will receive zero marks
💡When proving identities, always start with the more complex side and manipulate it to match the simpler side; never move terms across the equals sign as if solving an equation
💡For questions involving R cos(θ ± α), explicitly state the calculated values of R and α before substituting them back into the final equation
💡In 'Show that' questions, manipulate only one side of the identity (usually the LHS) to match the other; never move terms across the equals sign as this assumes the result
💡Always check the domain of the question immediately; if the range is given in terms of π, your answers must be in radians and exact forms
💡When solving equations involving k*x (e.g., sin 2x), adjust the interval first (e.g., 0 ≤ 2x ≤ 720) to ensure you capture all solutions before dividing back
💡In 'Show that' questions, you must document every algebraic step; jumping from the penultimate line to the answer without showing the intermediate simplification will lose the final accuracy mark
💡Always check the mode of your calculator; if the question involves calculus or small angle approximations, you must work in radians
💡When solving equations involving sec, cosec, or cot, convert to sin, cos, or tan immediately to minimise algebraic errors
💡For questions asking for 'exact values', do not use decimal approximations; leave answers in surd or pi form

Common Mistakes

Pitfalls to avoid in your exam answers

Dividing an equation by a common trigonometric factor (e.g., dividing by sin x) resulting in the loss of the solution sin x = 0
Failure to adjust the domain when solving for compound angles (e.g., solving for 2x + 30), leading to missing valid solutions within the original range
Using degree mode on calculators for calculus questions (differentiation/integration), which requires radians for validity
Incorrect signs when integrating trigonometric functions, specifically confusing the derivatives and integrals of sine and cosine
Dividing an equation by a trigonometric function (e.g., dividing by cos x) instead of factorizing, causing the loss of a valid set of solutions
Using degree mode calculators for questions involving calculus or small angle approximations, where radians are mandatory
Incorrectly assuming sin(A+B) = sinA + sinB, failing to apply the correct compound angle expansion
Omitting the constant of integration or misapplying signs when integrating trigonometric functions
Dividing an equation by a trigonometric function (e.g., dividing by cos x) resulting in the loss of solutions where the function equals zero; factorisation must be used instead
Confusing the signs in compound angle formulae, particularly incorrectly expanding cos(A - B) as cos A - cos B or cos A cos B - sin A sin B
Applying small angle approximations (e.g., sin x ≈ x) when the angle is in degrees; this technique is valid only in radians
Failure to adjust the interval when solving for transformed angles (e.g., solving for 2x + pi/3 but checking solutions against the original x interval)

Study Guide Available

Comprehensive revision notes & examples

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Key Terminology

Essential terms to know

Right-angled trigonometry (SOH CAH TOA)

Exact trigonometric values (0°, 30°, 45°, 60°, 90°)

Sine and Cosine Rules for non-right-angled triangles

Area of a triangle using ½absinC

Trigonometric graphs and periodicity

Three-dimensional trigonometric problems

Right-angled trigonometry (SOH CAH TOA)

Exact trigonometric values

The Sine Rule and Cosine Rule

Area of a triangle using 1/2 ab sin C

Trigonometric graphs and periodicity

Three-dimensional trigonometry

Bearings and elevation/depression

Right-angled ratios (SOH CAH TOA)

Sine and Cosine Rules for non-right-angled triangles

Trigonometric graphs and periodicity

Exact trigonometric values

3D geometric applications

Likely Command Words

How questions on this topic are typically asked

Solve

Show that

Prove

Determine

Find

Evaluate

Sketch

Find the exact value

Practical Links

Related required practicals

•{"code":"Modelling","title":"Simple Harmonic Motion","relevance":"Application of Rcos(θ-α) to analyze wave amplitude and phase shift in mechanics"}
•{"code":"Modelling","title":"Harmonic Motion","relevance":"Using R cos(theta - alpha) to analyse tidal heights or mechanical oscillations"}

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