Subject: Mathematics | Level: A-Level | Exam Board: OCR
Master OCR A-Level Trigonometry (4.9) with this comprehensive guide. We cover everything from reciprocal functions and compound angles to exam technique and common pitfalls, helping you secure top marks.
Revision Notes & Key Concepts
Revision Podcast Transcript
[INTRO - 1 minute] Hello and welcome to A-Level Maths Mastery! I'm your host, and today we're diving deep into one of the most powerful topics in OCR A-Level Mathematics: Trigonometry, specifically section 4.9. Whether you're preparing for your exams or just looking to strengthen your understanding, this episode will give you the tools you need to tackle those challenging trig questions with confidence. Trigonometry at A-Level goes way beyond SOHCAHTOA. We're talking about reciprocal functions like secant, cosecant, and cotangent, compound angle formulae, and the ability to prove complex identities. These skills are absolutely essential because they appear across pure maths, mechanics, and even in real-world applications like engineering and physics. So let's get started! [CORE CONCEPTS - 5 minutes] First, let's talk about the unit circle. This is your foundation for everything in trigonometry. The unit circle has a radius of one, and it maps every angle to a coordinate point. The x-coordinate gives you cosine, and the y-coordinate gives you sine. Understanding this relationship is crucial because it explains why sine and cosine have their specific ranges and why certain angles give exact values. Now, here's something that catches many candidates out: radians versus degrees. At A-Level, you must be fluent in both, but here's the key rule: if a question involves calculus or small angle approximations, you MUST work in radians. The OCR mark scheme is very strict on this. If you differentiate sine x and you're in degrees, you won't get the correct result. Radians are the natural unit for mathematical analysis. Let's move to reciprocal functions. Secant is one over cosine, cosecant is one over sine, and cotangent is one over tangent. These aren't just theoretical concepts; they appear frequently in exam questions. The key thing to remember is where these functions have asymptotes. Secant has vertical asymptotes wherever cosine equals zero, so that's at π over 2, 3π over 2, and so on. Cosecant has asymptotes where sine equals zero, which is at 0, π, 2π. Understanding the graphs of these functions and their behaviour is worth marks in the exam. Now, the Pythagorean identities. You already know sin²x plus cos²x equals 1, but at A-Level, you need the reciprocal versions too. Dividing through by cos²x gives you 1 plus tan²x equals sec²x. Dividing by sin²x gives you 1 plus cot²x equals cosec²x. These identities are your toolkit for solving equations and proving results. Examiners love to test whether you can spot which identity to use. Compound angle formulae are absolutely critical. Sin of A plus B is NOT sin A plus sin B. This is one of the most common mistakes. The correct formula is sin A cos B plus cos A sin B. For cosine, it's cos A cos B minus sin A sin B. And for tangent, it's tan A plus tan B over 1 minus tan A tan B. You must memorise these because they're not always given on the formula sheet, depending on the exam board. These formulae allow you to expand expressions, solve equations, and prove identities. Double angle formulae are just special cases where A equals B. So sin 2A is 2 sin A cos A. Cos 2A has three forms: cos²A minus sin²A, which can also be written as 2cos²A minus 1, or 1 minus 2sin²A. Knowing all three forms is powerful because different forms are useful in different contexts. [EXAM TIPS & COMMON MISTAKES - 2 minutes] Let's talk exam technique. In "show that" questions, you must manipulate only one side of the equation to match the other. Never move terms across the equals sign because that assumes what you're trying to prove. Start with the more complicated side and work towards the simpler side, showing every step clearly. When solving trigonometric equations, always check the domain first. If the question gives the range in terms of π, your answers must be in radians and in exact form. Don't give decimal approximations unless explicitly asked. And here's a critical point: never divide both sides of an equation by a trigonometric function. If you have sin x cos x equals sin x, don't divide by sin x. Instead, rearrange to sin x times cos x minus 1 equals zero, then factorise. Dividing loses solutions because you're assuming sin x isn't zero, but it might be! Another common error: calculator mode. If you're doing calculus or using small angle approximations like sin θ approximately equals θ for small θ, your calculator must be in radians. Degree mode will give completely wrong answers. For harmonic form questions, where you express a cos x plus b sin x as R cos of x minus α, make sure you state both R and α clearly, and check that α is in the correct quadrant. Marks are awarded for the correct value and the correct quadrant. [QUICK-FIRE RECALL QUIZ - 1 minute] Right, let's test your recall. I'll give you a few seconds after each question. Question 1: What is the exact value of sin 30 degrees? ... The answer is one half. Question 2: What is 1 plus tan²x equal to? ... It's sec²x. Question 3: What is the derivative of sin x? ... It's cos x, but only if x is in radians. Question 4: Where does sec x have vertical asymptotes? ... Wherever cos x equals zero, so at odd multiples of π over 2. Question 5: What is the compound angle formula for sin of A plus B? ... It's sin A cos B plus cos A sin B. How did you do? If you got them all, brilliant! If not, go back and review those concepts. [SUMMARY & SIGN-OFF - 1 minute] To wrap up, trigonometry at A-Level is all about fluency with identities, understanding the unit circle, and being meticulous with radians versus degrees. The examiners are looking for clear, logical working, correct use of identities, and all solutions within the given range. Remember: factorise, don't divide. Use exact values. Show every step in proofs. And always, always check your domain. Practice is key. Work through past papers, focus on the mark schemes, and understand why marks are awarded. Trigonometry is a skill that improves dramatically with practice, and once it clicks, you'll find it's one of the most satisfying topics in A-Level maths. Thanks so much for listening to A-Level Maths Mastery. Keep practising, stay confident, and I'll see you in the next episode. Good luck with your exams!
Key Terms & Definitions
- Radian
- The angle subtended at the center of a circle by an arc that is equal in length to the radius.
- Identity
- An equation that is true for all values of the variables for which both sides are defined.
- Asymptote
- A line that a curve approaches but never touches.
- Principal Value
- The unique solution to an inverse trigonometric function within a restricted range (e.g., -π/2 ≤ arcsin(x) ≤ π/2).
- Harmonic Form
- The process of expressing a sum of sine and cosine functions, a.cosx + b.sinx, as a single cosine or sine function, Rcos(x-α) or Rsin(x+α).
- Period
- The interval over which a periodic function completes one full cycle.
Worked Examples
Worked Example
Question: Solve the equation 2cos²x + sinx = 1 for the interval 0 ≤ x < 2π. Give your answers in terms of π.
Solution: Step 1: The equation contains both cos and sin terms. To solve it, we need to express it in terms of a single trigonometric function. Use the identity sin²x + cos²x = 1, which rearranges to cos²x = 1 - sin²x. Step 2: Substitute this into the equation: 2(1 - sin²x) + sinx = 1. Step 3: Expand and simplify the equation: 2 - 2sin²x + sinx = 1. This gives a quadratic in sinx: 2sin²x - sinx - 1 = 0. Step 4: Let y = sinx. The equation becomes 2y² - y - 1 = 0. Factorise the quadratic: (2y + 1)(y - 1) = 0. Step 5: Solve for y: y = -1/2 or y = 1. This means sinx = -1/2 or sinx = 1. Step 6: Find the principal value for each case. For sinx = 1, x = π/2. For sinx = -1/2, the principal value is x = -π/6. Step 7: Find all solutions in the given range 0 ≤ x < 2π. For sinx = 1, x = π/2 is the only solution. For sinx = -1/2, the solutions are in the 3rd and 4th quadrants. x = π + π/6 = 7π/6 and x = 2π - π/6 = 11π/6. Final answer: x = π/2, 7π/6, 11π/6.
Worked Example
Question: Show that the equation tan(x + 45°) = 2cotx can be written as tan²x + 3tanx - 1 = 0.
Solution: Step 1: Start with the left-hand side (LHS) and apply the compound angle formula for tan(A+B): tan(x + 45°) = (tanx + tan45°)/(1 - tanxtan45°). Step 2: We know that tan45° = 1. Substitute this value into the expression: (tanx + 1)/(1 - tanx). Step 3: Now, set this equal to the right-hand side (RHS), which is 2cotx. Also, replace cotx with 1/tanx: (tanx + 1)/(1 - tanx) = 2/tanx. Step 4: Multiply both sides by tanx and (1 - tanx) to eliminate the denominators: tanx(tanx + 1) = 2(1 - tanx). Step 5: Expand both sides: tan²x + tanx = 2 - 2tanx. Step 6: Rearrange the terms to form the required quadratic equation: tan²x + 3tanx - 1 = 0. This matches the required form.
Worked Example
Question: Express 3cosx + 4sinx in the form Rcos(x - α), where R > 0 and 0 < α < π/2. Hence, find the maximum value of the expression and the smallest positive value of x for which it occurs.
Solution: Step 1: Use the Rcos(x - α) expansion: Rcos(x - α) = R(cosxcosα + sinxsinα) = (Rcosα)cosx + (Rsinα)sinx. Step 2: Compare coefficients with 3cosx + 4sinx. We have Rcosα = 3 and Rsinα = 4. Step 3: To find R, square and add the two equations: (Rcosα)² + (Rsinα)² = 3² + 4². This gives R²(cos²α + sin²α) = 9 + 16 = 25. Since R > 0, R = 5. Step 4: To find α, divide the two equations: (Rsinα)/(Rcosα) = tanα = 4/3. So, α = arctan(4/3) ≈ 0.927 radians (to 3 s.f.). Step 5: The expression is 5cos(x - 0.927). Step 6: The maximum value of the cosine function is 1. Therefore, the maximum value of the expression is 5 * 1 = 5. Step 7: This maximum occurs when cos(x - 0.927) = 1. The principal value is x - 0.927 = 0. Therefore, the smallest positive value of x is x = 0.927 radians.
Practice Questions
Question: Solve the equation 3sec²x - 4tanx - 2 = 0 for 0 ≤ x ≤ 180°.
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Question: Find the exact value of sin(15°).
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Question: Prove the identity: (sinx + sin2x) / (1 + cosx + cos2x) ≡ tanx.
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Question: A voltage V is modelled by the equation V = 120cos(100t) + 160sin(100t). Express V in the form Rcos(100t - α), where R > 0 and 0 < α < 90°. State the maximum voltage.
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Question: Given that sin(x) = -5/13 and x is an obtuse angle in the third quadrant, find the exact value of cos(x) and tan(x).
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