Ch 5 · 1.5 Trigonometry · A Level Mathematics 9709
Topic 5 of 8 in Pure Mathematics 1 — A Level Mathematics 9709 — 73 lessons.
Introduction
Tides, sound waves, alternating current and the height of a seat on a big wheel all go up and down in a repeating pattern described by sine and cosine. In Paper 1 you sketch these waves, use exact values, prove identities, and solve equations, finding every solution in the interval the question gives. (Syllabus section 1.5, Paper 1.)
Key ideas — The three graphs
Transformed graphs use section 1.2: y = 3sin x has amplitude 3; y = sin 2x has period 180^°; y = 1 - cos 2x is cos x stretched by factor tfrac12 along x, reflected in the x-axis and translated up 1, so it has period π and runs from 0 to 2; y = tan (x + π/4 ) is tan x translated π/4 to the left.
🧑🏫 Teacher's explanation
Think of trigonometric functions as repetitive patterns, like the rising and falling of a tide or the oscillation of a pendulum. The sine and cosine functions are 'waves' that smoothly oscillate between a maximum and minimum value. If you imagine a point moving around a circle, the sine function tracks its vertical height, while the cosine tracks its horizontal position. Because the circle repeats every 360^° (or 2π radians), the graphs repeat their shape forever.
📖 Definitions
🔍 Worked example — Easy
Sketch the graph of y = 2sin(x) for 0 ≤ x ≤ 2π.
🔍 Worked example — Standard
Sketch y = 1 - cos(2x) for 0 ≤ x ≤ π.
🔍 Worked example — Exam-style
Solve tan(x) = 1 for 0 ≤ x ≤ \2π.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Key ideas — Exact values
Related angles use the symmetry of the graphs: sin 150^° = sin 30^° = tfrac12, cos4π/3 = -cosπ/3 = -tfrac12, tan 135^° = -1.
🧑🏫 Teacher's explanation
In trigonometry, we often deal with angles that don't result in clean, terminating decimals like 0.5 or 0.707. Instead, they result in 'exact values'—fractions involving square roots. These values come from two special right-angled triangles: the 45^°-45^°-90^° triangle and the 30^°-60^°-90^° triangle. Think of these exact values like a 'cheat sheet' for the unit circle; instead of using a calculator to approximate sin(60^°) as 0.866, we use the precise value √(3)/2.
📖 Definitions
🔍 Worked example — Easy
Evaluate cos(120^°) using exact values.
🔍 Worked example — Standard
Find the exact value of tan(4π/3).
🔍 Worked example — Exam-style
Solve sin(x) = -√(3)/2 for 0^° ≤ x < 360^°.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Key ideas — Principal values
sin x = 0.5 has infinitely many solutions; the calculator gives one, the principal value:
🧑🏫 Teacher's explanation
Imagine you are looking at a Ferris wheel. If I tell you the height of a passenger is 0.5 meters above the center, there are actually two different positions on the wheel where this happens: once while the wheel is going up, and once while it is going down. Because the wheel keeps spinning, this happens every single rotation, meaning there are infinitely many answers.
📖 Definitions
🔍 Worked example — Easy
Find the principal value of sin^-1(0.5) in degrees.
🔍 Worked example — Standard
Find the principal value of cos^-1(-1/2) in degrees.
🔍 Worked example — Exam-style
Solve tan x = -1 for 0^° ≤ x < 360^°.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Key ideas — Two identities
tanθ ≡ sinθ/cosθ sin^2θ + cos^2θ ≡ 1
🧑🏫 Teacher's explanation
Think of these two identities as the 'universal translators' of trigonometry. Just as you might convert currency to a common base to compare prices, these identities allow you to convert between different trigonometric ratios so they can 'talk' to each other.
📖 Definitions
🔍 Worked example — Easy
Prove the identity tanθ · cosθ = sinθ.
🔍 Worked example — Standard
Solve cos^2 x = 0.6 for 0^° ≤ x ≤ 90^°.
🔍 Worked example — Exam-style
Show that 1/sin^2θ - 1/tan^2θ = 1.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Key ideas — Finding all solutions in an interval
Equations with both sin^2 and cos (or cos^2 and sin) become quadratics once you use sin^2θ + cos^2θ ≡ 1.
🧑🏫 Teacher's explanation
When you solve an equation like sin(x) = 0.5, your calculator only gives you one answer, called the 'principal value'. However, trigonometric functions are periodic, meaning they repeat their values over and over again. Imagine a Ferris wheel: if you are at a height of 5 meters, you will reach that height twice during every single rotation—once while going up and once while going down.
📖 Definitions
🔍 Worked example — Easy
Solve sin(x) = 1/2 for 0^° ≤ x < 360^°.
🔍 Worked example — Standard
Solve cos(2x) = -0.5 for 0^≤ x < 2π.
🔍 Worked example — Exam-style
Solve 2cos^2 x + sin x - 1 = 0 for 0^° ≤ x < 360^°.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Worked examples — Example 1: a quadratic in cos
Solve 2sin^2θ + 3cosθ = 0 for 0^° ≤ θ ≤ 360^°.
🔍 Worked example — Easy
Solve 2cos^2θ - cosθ - 1 = 0 for 0^° ≤ θ ≤ 360^°.
🔍 Worked example — Standard
Solve 2sin^2θ + 3cosθ = 0 for 0^° ≤ θ ≤ 360^°.
🔍 Worked example — Exam-style
Solve 3cos^2θ + sinθ - 1 = 0 for 0^° ≤ θ ≤ 360^°.
⚠️ Common errors
Worked examples — Example 2: a multiple angle in radians
Solve sin 2x = -tfrac12 for 0 ≤ x ≤ π.
🔍 Worked example — Easy
Solve cos 2x = 1 for 0 ≤ x ≤ π.
🔍 Worked example — Standard
Solve sin 2x = -1/2 for 0 ≤ x ≤ π.
🔍 Worked example — Exam-style
Solve cos 3x = 1/2 for 0 ≤ x ≤ π.
⚠️ Common errors
Worked examples — Example 3: a quadratic in tan
Solve tan^2 x = 1 + tan x for 0^° ≤ x ≤ 360^°.
🔍 Worked example — Easy
Solve 2tan^2 x - tan x - 1 = 0 for 0^° ≤ x < 180^°.
🔍 Worked example — Standard
Solve tan^2 x = 1 + tan x for 0^° ≤ x < 360^°.
🔍 Worked example — Exam-style
Solve tan^2 x - 3 = 0 for 0 ≤ x < 2π (radians).
⚠️ Common errors
Worked examples — Example 4: proving an identity
Prove that 1/cosθ - cosθ ≡ sinθtanθ.
🔍 Worked example — Easy
Prove that 1/secθ ≡ cosθ
🔍 Worked example — Standard
Prove that sinθ/1 + cosθ + 1 + cosθ/sinθ ≡ 2/sinθ
🔍 Worked example — Exam-style
Prove that 1 - cosθ/sinθ ≡ sinθ/1 + cosθ
⚠️ Common errors
Common mistakes
Practice
Answers: (1) sqrt3/2, -sqrt3. (2) period 180^° (π); range -1 ≤ y ≤ 5. (3) sinθ (1/cosθ - 3 ) = 0: sinθ = 0 ⇒ 0^°, 180^°, 360^°; cosθ = tfrac13 ⇒ 70.5^°, 289.5^°. (4) x - 30^° = 66.4^° or 293.6^°, so x = 96.4^°, 323.6^°. (5) 1 - cos^2θ/1 - cosθ = (1-cosθ)(1+cosθ)/1 - cosθ = 1 + cosθ. (6) 3sin^2 x + sin x - 2 = 0 ⇒ (3sin x - 2)(sin x + 1) = 0; sin x = tfrac23: x = 0.730, 2.41; sin x = -1: x = 3π/2 = 4.71.