Ch 3 · 2.3 / 3.3 Trigonometry · A Level Mathematics 9709
Topic 3 of 9 in Pure Mathematics 2 and 3 — A Level Mathematics 9709 — 72 lessons.
Introduction
Paper 1 used sin, cos, tan and two identities. Now the toolkit grows: three reciprocal functions, compound-angle and double-angle formulae, and the "R form" that turns 3sinθ + 4cosθ into a single wave. Engineers use exactly this to combine two alternating signals. The content is identical for Paper 2 (section 2.3) and Paper 3 (section 3.3).
Key ideas — Reciprocal functions
secθ = 1/cosθ cscθ = 1/sinθ cotθ = 1/tanθ = cosθ/sinθ
🧑🏫 Teacher's explanation
Think of reciprocal functions as the 'flipped' versions of the standard trigonometric functions. If sinθ is a fraction like 1/2, then its reciprocal cscθ is simply the flipped value, 2/1. This means that wherever a standard function is very small (close to zero), its reciprocal becomes extremely large, shooting off towards infinity.
📖 Definitions
🔍 Worked example — Easy
Show that sec^2θ - tan^2θ = 1 for all values of θ where the functions are defined.
🔍 Worked example — Standard
Solve the equation csc^2θ = 4 for 0^° ≤ θ ≤ 360^°.
🔍 Worked example — Exam-style
Solve sec^2θ + tanθ = 3 for 0 ≤ θ < 2π.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Key ideas — Pythagorean identities
Divide sin^2θ + cos^2θ ≡ 1 by cos^2θ and by sin^2θ:
🧑🏫 Teacher's explanation
Think of trigonometric identities as 'exchange rates' in a currency market. Just as you can swap 1 US Dollar for 100 Japanese Yen, you can swap sin^2θ for 1 - cos^2θ. The Pythagorean identity sin^2θ + cos^2θ = 1 is the fundamental rule that governs how these 'currencies' relate to one another.
📖 Definitions
🔍 Worked example — Easy
Simplify the expression 1/cos^2θ - tan^2θ.
🔍 Worked example — Standard
Solve the equation 2sec^2θ + 3tanθ = 0 for 0^° ≤ θ ≤ 360^°.
🔍 Worked example — Exam-style
Solve csc^2θ - cot^2θ = k for k.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Key ideas — Compound-angle formulae
sin(A ± B) ≡ sin Acos B ± cos Asin B cos(A ± B) ≡ cos Acos B ∓ sin Asin B tan(A ± B) ≡ tan A ± tan B/1 ∓ tan Atan B
🧑🏫 Teacher's explanation
Imagine you are trying to find the exact coordinates of a point on a circle, but the angle you are looking at is a 'messy' one, like 75^° or 15^°. You don't have these on your standard unit circle, but you do know 30^°, 45^°, and 60^°. Compound angle formulae act like a 'recipe' that lets you break down a complex angle into a combination of these simple, known angles.
📖 Definitions
🔍 Worked example — Easy
Use compound angle formulae to find the exact value of cos(15^°).
🔍 Worked example — Standard
Show that sin(2θ) = 2sinθcosθ.
🔍 Worked example — Exam-style
Solve the equation cos(2x) = 1/2 for 0^° ≤ x ≤ 360^°.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
(b) Solve the equation cos 2θ = sin θ for 0^≤ θ ≤ 360^°, giving your answers to the nearest degree. [4] [7 marks]
Key ideas — Double-angle formulae
sin 2A ≡ 2sin Acos A tan 2A ≡ 2tan A/1 - tan^2 A cos 2A ≡ cos^2 A - sin^2 A ≡ 2cos^2 A - 1 ≡ 1 - 2sin^2 A
🧑🏫 Teacher's explanation
Imagine you are looking at a wave. If you double the frequency of that wave, the wave repeats twice as fast. Double-angle formulae allow us to translate between an angle and its double, which is essential when we need to change the 'speed' of a trigonometric function to solve an equation or perform calculus.
📖 Definitions
🔍 Worked example — Easy
Use the double-angle formula for sin 2A to find the exact value of sin 15^° given that cos 15^° = √(6)+√(2)/4.
🔍 Worked example — Standard
Solve the equation cos 2θ = cos θ for 0^° ≤ θ ≤ 360^°.
🔍 Worked example — Exam-style
Show that 1 - cos 2θ/sin 2θ = tan θ.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Key ideas — The R form
For positive a and b:
🧑🏫 Teacher's explanation
Imagine you are trying to combine two different waves—one a sine wave and one a cosine wave—into a single, simpler wave. If you have asinθ + bcosθ, it looks messy because you have two different functions. However, because sine and cosine are just shifted versions of each other, their sum is actually just another single sine wave, just with a new amplitude R and a new starting phase α.
📖 Definitions
🔍 Worked example — Standard
Solve 5sinθ + 12cosθ = 13 for 0^° ≤ θ ≤ 360^\°.
🔍 Worked example — Exam-style
Find the maximum and minimum values of f(θ) = √(3)sinθ - cosθ.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Worked examples — Example 1: a quadratic in tan
Solve 2sec^2θ - 5tanθ = 0 for 0^° ≤ θ ≤ 360^°.
🔍 Worked example — Easy
Solve tan^2θ - 3 = 0 for 0^° ≤ θ ≤ 180^°.
🔍 Worked example — Standard
Solve 2sec^2θ - 5tanθ = 0 for 0^° ≤ θ ≤ 360^°.
🔍 Worked example — Exam-style
Solve sec^2θ + tanθ = 1 for 0^≤ θ ≤ π.
⚠️ Common errors
Worked examples — Example 2: using a double angle
Solve tanθ + cotθ = 4 for 0^° < θ < 360^°.
🔍 Worked example — Easy
Solve tanθ = 1 for 0^° ≤ θ ≤ 360^°.
🔍 Worked example — Standard
Solve tanθ + cotθ = 4 for 0^° < θ < 360^°.
🔍 Worked example — Exam-style
Solve cos 2θ = -1/2 for 0^° ≤ θ < 180^°.
⚠️ Common errors
Worked examples — Example 3: a compound angle
Solve tan(θ + 45^°) = 6tanθ for 0^° < θ < 180^°.
🔍 Worked example — Easy
Use the expansion of tan(A + B) to simplify tan(θ + 45^°) in terms of tanθ.
🔍 Worked example — Standard
Solve tan(θ + 45^°) = 6tanθ for 0^° < θ < 180^°.
🔍 Worked example — Exam-style
Solve cos(x - 30^°) = 3sin(x - 60^°) for 0^° < x < 180^°.
⚠️ Common errors
Worked examples — Example 4: the R form
Express 3sinθ + 4cosθ as Rsin(θ + α) and solve 3sinθ + 4cosθ = 2 for 0^° ≤ θ ≤ 360^°.
🔍 Worked example — Easy
Express 3sinθ + 4cosθ in the form Rsin(θ + α), where α > 0.
🔍 Worked example — Standard
Solve 3sinθ + 4cosθ = 2 for 0^° ≤ θ ≤ 360^°.
🔍 Worked example — Exam-style
Find the maximum value of f(θ) = √(3)sinθ - cosθ.
⚠️ Common errors
Common mistakes
Practice
Answers: (1) sin(45^° - 30^°) = sqrt6 - sqrt2/4. (2) 3 + 3cot^2θ = 5 - cotθ ⇒ (3cotθ - 2)(cotθ + 1) = 0: tanθ = tfrac32 ⇒ 56.3^°; tanθ = -1 ⇒ 135^°. (3) 2sin^2θ/2sinθcosθ = sinθ/cosθ = tanθ. (4) cos A = tfrac45: sin 2A = 24/25, cos 2A = 1 - 2 (9/25 ) = 7/25. (5) R = √(29), tanα = tfrac52, α = 68.2^°: √(29)cos(x + 68.2^°); least value -√(29).