For Integrated Programme students: Your current school materials, teacher instructions, and assessment scope take precedence because IP topic sequence and depth vary by school. This is an Eclat IP guide, not the O-Level / SEC G3 exam-track guide.
How this chapter applies
Eclat core: six trigonometric functions, angles in degrees or radians, principal inverse values, exact values, graphs, periodicity, symmetry, and basic identities form the main route.
School-sensitive extension: inequality proofs, sector applications, and more abstract identity manipulation may vary by school.
2027 national comparison: K341 Topic G1 covers six functions for any angle, inverse principal values, exact values, transformed graphs, and core quotient, reciprocal, and Pythagorean identities.
Check your school: confirm graph-domain conventions, exact-angle sets, and whether general solutions are excluded.
Q: What does IP AMaths Notes (Upper Sec, Year 3-4): 12) Trigonometry I cover? A: Radian measure, special angles, and core identities for Additional Mathematics trigonometry.
Everything in upper-sec trig is built on radians and the primary identities. Memorise the exact values and practise expressing angles in coterminal forms.
Keep the full topic roadmap handy via our IP Maths tuition hub so you can jump into related drills, quizzes, or diagnostics as you move through these notes.
The core idea is simple: Trigonometry I is radians, exact values, and the core identities.
Use it as a working check: Convert angles into radians early, keep exact values when possible, and use identities to simplify before substituting numbers.
Then go one layer deeper: Example: for sector questions, use s = r theta and A = 1/2 r^2 theta, with theta in radians. Degrees must be converted first.
Choosing the trigonometry route
Most early AMaths trigonometry errors come from starting with the wrong form of the angle. Decide whether the question is about length, area, exact values, or algebra before substituting numbers.
Question cue
First move
Why it works
Arc length, sector area, or angular motion
Convert the angle to radians, then use s=rθ or A=21r2θ.
The sector formulae only work directly when θ is in radians.
Exact value such as sin210∘
Find the reference angle and quadrant first.
The quadrant gives the sign, while the reference angle gives the exact magnitude.
Expression with sin2θ, cos2θ, or tan2θ
Expression with 2π−θ
Use co-function identities.
Complementary angles swap sine and cosine.
Common trap: degrees and radians can both describe the same angle, but they are not interchangeable inside formulas. Convert once, write the unit beside the converted angle, and keep that form for the rest of the solution.
Exact-value sign checkpoint
For angles outside the first quadrant, split the job into sign and size before writing the final value.
Step
What to do
Example for 210∘
1
Locate the quadrant.
210∘ is in quadrant III.
2
Find the reference angle.
210∘−180∘=30∘.
3
Decide the sign using ASTC.
In quadrant III, sine and cosine are negative, tangent is positive.
4
Attach the exact value.
sin210∘=−21, cos210∘=−23
Misconception check: the reference angle gives only the magnitude. The quadrant decides the sign.
1 Key facts
Radian definition: arc length s=rθ.
Area of sector: A=21r2θ.
Identities: sin2θ+cos2θ=1; 1+tan2θ=sec2θ
Co-function: sin(2π−θ)=cosθ, cos(2π−θ)=sinθ
Six-functions and graph checkpoint
The six functions are sinx, cosx, tanx, cscx=1/sinx, secx=1/cosx, and cotx=1/tanx. Their reciprocal forms explain where each function is undefined.
For inverse trigonometric functions, the calculator returns a principal value. Treat it as a reference point, then use signs and the required interval to find all valid angles. For transformed graphs such as y=asin(bx)+c: ∣a∣ gives the amplitude, 2π/∣b∣ gives the period, and c shifts the midline. For y=atan(bx)+c, the period is π/∣b∣ and there is no amplitude.
Misconception check: sin−1x means the inverse sine function, while 1/sinx is cscx. They are not the same operation.
Identity route checkpoint
Before expanding a trigonometric expression, look for the shape of the terms. A good identity choice usually reduces the number of functions or turns a square pair into 1.
Expression shape
First identity to try
Why it helps
Common trap
sin2θ+cos2θ, or a close rearrangement
Pythagorean identity
Replace the pair with 1, or rearrange until only one function remains.
Expanding squared brackets before checking whether the pair is already present.
1+tan2θ, sec2θ−1, or a tangent-sec square pair
1+tan2θ=sec2θ
A mixture of tanθ, cotθ, sinθ, and cosθ
Quotient and reciprocal identities
Express everything in sine and cosine before combining fractions.
Cancelling terms from a sum before there is a common factor.
2π−θ or 90∘−θ
Worked check: tanθ+cotθ=cosθsinθ+sinθcosθ=sinθcosθsin2θ+cos2θ=sinθcosθ1.
Misconception check: identities are replacement rules under their valid domains. They do not let you cancel a term from a sum unless that term is a common factor of the whole numerator and denominator.
Arc-sector formula checkpoint
Before using a sector formula, identify whether the answer is a length, an area, or an angle. Then check that the angle is in radians.
Question asks for
Use
Unit check
Common trap
Arc length
s=rθ
Length unit, such as cm or m.
Using 21r2θ just because the diagram is a sector.
Sector area
A=21r2θ
Area unit, such as cm2
Angle from arc length
θ=rs
Radians have no length unit after division.
Leaving the answer in degrees without converting if the rest of the question uses radians.
Angle from sector area
θ=r22A
Area divided by radius squared leaves radians.
Dividing by r
Worked check: if r=6cm and θ=3π, arc length is 6×3π=2π,cm, while sector area is 21×62×3π=6π,cm2. The same angle appears in both formulae, but the units tell you which answer you found.
Misconception check: radians are required for these compact formulae. If the question gives degrees, convert first and do not mix the two angle forms in the same substitution.
2 Worked example - Arc and sector
A circle of radius 8cm subtends an angle of 125π. Find the arc length and sector area.
Arc length: s=rθ=8×125π=310πcm.
Area: A=21r2θ=21×64×125π=1280π=320πcm2
3 Worked example - Identity manipulation
Show that tanθ+cotθ≥2 for 0<θ<2π.
Express tanθ+cotθ=cosθsinθ+sinθcosθ.
Combine: sinθcosθsin2θ+cos2θ=sinθcosθ1
Using double-angle identity, sinθcosθ=21sin2θ.
Hence tanθ+cotθ=sin2θ2.
For 0<θ<2π, 0<2θ<π
Therefore expression is ≥2 with equality when sin2θ=1 i.e. θ=4π
4 Worked example - Radian conversion in context
A wheel of radius 0.35m turns through 150∘.
Find the angular displacement in radians, the arc length traced on the rim, and the area swept by the corresponding sector.
