Ch 2 · 10 Trigonometry · O Level Additional Mathematics 4037
Topic 2 of 2 in Trigonometry — O Level Additional Mathematics 4037 — 49 lessons.
Introduction
The height of a point on a turning wheel, the water level at a harbour, the voltage of mains electricity: all follow sine and cosine waves. This topic extends trigonometry to any angle, introduces three reciprocal functions, and uses identities to solve equations and prove results. (Syllabus topic 10.)
Key ideas — Six functions of any angle
sec x = 1/cos x csc x = 1/sin x cot x = 1/tan x = cos x/sin x
🧑🏫 Teacher's explanation
In right-angled trigonometry, you only deal with acute angles (between 0^° and 90^°). But in Add Maths, we treat trigonometric functions as 'machines' that can take any number—positive, negative, or massive—and spit out a ratio. Think of the six functions as a family of six different lenses. The primary three (sin, cos, tan) are the foundation. The other three (csc, sec, cot) are just their 'reciprocals'—mathematical mirrors that flip the ratio upside down.
📖 Definitions
🔍 Worked example — Easy
Given that sin θ = 0.6 and θ is in the second quadrant, find the exact value of csc θ.
🔍 Worked example — Standard
Prove that tan x · csc x = sec x.
🔍 Worked example — Exam-style
Solve sec^2 θ = 4 for 0^° ≤ θ ≤ 360^°.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Key ideas — Amplitude and period
For y = asin bx + c and y = acos bx + c (with a > 0):
🧑🏫 Teacher's explanation
Think of a trigonometric graph like a wave traveling along a beach. The 'amplitude' is how high the wave peaks and how low the troughs go relative to the calm water level. If the water level is at c, and the wave height is a, the wave reaches up to c+a and down to c-a.
📖 Definitions
🔍 Worked example — Easy
State the amplitude, period, maximum value, and minimum value of y = 4cos(3x) + 2.
🔍 Worked example — Standard
Find the period and the x-coordinates of the asymptotes for y = tan(1/2x) - 3 for 0^° ≤ x < 360^°.
🔍 Worked example — Exam-style
A trigonometric graph is given by y = asin(bx) + c. The maximum value is 7, the minimum value is -3, and the period is 90^°. Find the values of a, b, and c.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Key ideas — Identities
sin^2 A + cos^2 A = 1 sec^2 A = 1 + tan^2 A csc^2 A = 1 + cot^2 A
🧑🏫 Teacher's explanation
Trigonometric identities are like 'exchange rates' in currency. Just as you can swap 100 cents for 1 dollar, these identities allow you to swap one trigonometric function for another without changing the value of the expression. This is vital when you are faced with an equation containing two different functions, like sin x and cos x, and you need them to 'speak the same language' to solve for x.
📖 Definitions
🔍 Worked example — Easy
Show that cos^2 θ = 1 - sin^2 θ.
🔍 Worked example — Standard
Solve 2sin^2 x + 3cos x - 3 = 0 for 0^° ≤ x ≤ 360^°.
🔍 Worked example — Exam-style
Prove that tan A/sec A = sin A.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Key ideas — Solving equations in a given domain
Never divide by a function that could be zero; factorise instead.
🧑🏫 Teacher's explanation
Solving trigonometric equations is like finding all the points where a wave hits a certain height. Because waves repeat, there isn't just one answer; there are often multiple answers within a specific range (the domain). Think of a Ferris wheel: if you want to know when you are exactly 5 meters above the ground, you will hit that height twice during one full rotation—once going up and once going down.
📖 Definitions
🔍 Worked example — Easy
Solve 2sin θ = 1 for 0^° ≤ θ ≤ 360^°.
🔍 Worked example — Standard
Solve 2cos^2 x + sin x - 1 = 0 for 0 ≤ x ≤ 2π.
🔍 Worked example — Exam-style
Solve tan(2 heta) = 1 for 0^° ≤ θ ≤ 180^°.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Key ideas — Proving identities
Start with the more complicated side, rewrite everything in sin and cos, combine fractions, and use the identities until you reach the other side. Do not work on both sides at once as if solving an equation.
Worked examples
Solve 4cotθ = tanθ for 0^° ≤ θ ≤ 360^°. tan^2θ = 4 ⇒ tanθ = ± 2: θ = 63.4^°, 116.6^°, 243.4^°, 296.6^°.
🔍 Worked example — Easy
Solve 4cotθ = tanθ for 0^° ≤ θ ≤ 360^°.
🔍 Worked example — Standard
Solve 2sec^2θ + tanθ - 3 = 0 for 0^° ≤ θ ≤ 360^°.
🔍 Worked example — Exam-style
Prove that sinθ/1 + cosθ + 1 + cosθ/sinθ = 2cscθ.
⚠️ Common errors
Common mistakes
Practice
Answers: (1) amplitude 4, period 120^°; greatest 2, least -6. (2) sin 2x = tfrac34, 0^° ≤ 2x ≤ 360^°: 2x = 48.6^°, 131.4^°, so x = 24.3^°, 65.7^°. (3) 2(1 - sin^2 x) = 3sin x ⇒ (2sin x - 1)(sin x + 2) = 0: sin x = tfrac12, x = 30^°, 150^°. (4) sin^2 x/cos x + cos x = sin^2 x + cos^2 x/cos x = 1/cos x = sec x. (5) cosθ (1/sinθ - 3 ) = 0: cosθ = 0 ⇒ π/2; sinθ = tfrac13 ⇒ 0.340, 2.80.