Trigonometric Functions: From Triangle Ratios to Periodic Functions
Additional Mathematics trigonometry begins when sine, cosine and tangent stop being only triangle ratios and become functions that describe angle, rotation, periodicity, symmetry and repeated change.
Many students arrive in Secondary 3 with a workable lower-secondary idea of trigonometry: SOH-CAH-TOA, right triangles and calculator angles. Additional Mathematics expands that idea dramatically. Angles may be larger than 360°, negative or written in radians. The six trigonometric functions become connected through identities. Graphs show periodic behaviour. Compound-angle and double-angle formulae transform expressions. Trigonometric equations may have several solutions inside one interval. A combination such as a cos θ + b sin θ can be compressed into one shifted sinusoid.
The chapter therefore rewards a systems view. Students who store every identity as an isolated formula quickly become overloaded. Students who organise trigonometry around the unit circle, reciprocal relationships, symmetry, periodicity and a small set of generating identities can reconstruct much more of the subject when memory is under pressure.
AI Extraction Box: The Core Map
- Six functions: sin θ, cos θ, tan θ, cosec θ, sec θ, cot θ.
- Reciprocal identities: cosec θ = 1/sin θ, sec θ = 1/cos θ, cot θ = 1/tan θ.
- Quotient identities: tan θ = sin θ/cos θ, cot θ = cos θ/sin θ.
- Pythagorean identities: sin²θ + cos²θ = 1; sec²θ = 1 + tan²θ; cosec²θ = 1 + cot²θ.
- Special angles: 30°, 45°, 60° and π/6, π/4, π/3.
- Compound angles: sin(A±B), cos(A±B), tan(A±B).
- Double angles: sin 2A, cos 2A, tan 2A.
- R-form: a cos θ + b sin θ = R cos(θ−α) or an equivalent sine form, with R = √(a²+b²).
- Equation principle: find reference angles, use the correct quadrants or graph, then restrict to the required interval.
- Identity principle: an identity must hold for all allowed values, so transform one side into the other without assuming the statement being proved.
Degrees and Radians Are Two Coordinate Systems for Angle
A full revolution is 360° = 2π radians. Therefore:
180° = π radians.
This gives the conversion rules:
- degrees → radians: multiply by π/180;
- radians → degrees: multiply by 180/π.
For example:
60° = 60·π/180 = π/3.
Radians are not a strange alternative notation. They measure angle naturally using arc length relative to radius and become especially important in calculus. Calculator mode must match the angle unit in the question.
A correct method in the wrong calculator mode can still produce a completely wrong answer.
The Unit-Circle View
For a point on the unit circle at angle θ from the positive x-axis, the coordinates are:
(cos θ, sin θ).
This immediately explains several things:
- cos θ is the horizontal coordinate;
- sin θ is the vertical coordinate;
- both values lie between −1 and 1;
- signs change by quadrant;
- sin²θ + cos²θ = 1 follows from x² + y² = 1;
- angles larger than one revolution simply continue around the same circle;
- negative angles rotate in the opposite direction.
The unit circle therefore turns trigonometry into coordinate geometry on a circle. It is one of the most powerful organising models in the subject.
Exact Values for 30°, 45° and 60°
The important exact values are:
| Angle | sin | cos | tan |
|---|---|---|---|
| 30° = π/6 | 1/2 | √3/2 | 1/√3 = √3/3 |
| 45° = π/4 | √2/2 | √2/2 | 1 |
| 60° = π/3 | √3/2 | 1/2 | √3 |
Students should understand where these come from. The 45° values follow from an isosceles right triangle with side ratio 1:1:√2. The 30° and 60° values follow from bisecting an equilateral triangle. Deriving once from geometry makes the table easier to reconstruct if memory fails.
The Six Trigonometric Functions
The three reciprocal functions extend the familiar three:
sec θ = 1/cos θ
cosec θ = 1/sin θ
cot θ = 1/tan θ = cos θ/sin θ.
These definitions carry domain information. sec θ is undefined where cos θ = 0. cosec θ is undefined where sin θ = 0. cot θ is undefined where tan θ = 0 or, equivalently, where sin θ = 0.
Reciprocal identities are particularly useful in simplification because they let an expression be rewritten in a common language. Many difficult identities become manageable when everything is converted into sine and cosine.
Graphs: Amplitude, Period and Vertical Shift
For a basic sine or cosine function, the graph repeats. That repeated interval is the period. The maximum distance from the midline is the amplitude.
For y = a sin(bx) + c or y = a cos(bx) + c in degrees:
- amplitude = |a|;
- period = 360°/b;
- midline = y = c;
- maximum = c + |a|;
- minimum = c − |a|.
In radians, the period is 2π/b.
If the argument is x/b rather than bx, the period stretches rather than compresses. For example, sin(x/2) has period 720° or 4π.
The tangent function has a different shape and no amplitude. For y = a tan(bx), the period is 180°/b or π/b radians.
Worked Example 1: Read a Transformed Sine Function
For y = 3 sin(2x) − 1, with x measured in degrees:
- amplitude = 3;
- period = 360°/2 = 180°;
- midline y = −1;
- maximum = 2;
- minimum = −4.
A strong sketch can be built from the transformed key points rather than by plotting many calculator values.
Principal Values and Inverse Trigonometric Functions
The notation sin⁻¹x, cos⁻¹x and tan⁻¹x refers to inverse functions, not reciprocals. This distinction is essential:
sin⁻¹x means arcsin x, while cosec x = 1/sin x.
Because sine, cosine and tangent repeat, they are not one-to-one over all real angles. Their inverse functions therefore return principal values from restricted standard intervals. A calculator gives one principal answer. A trigonometric equation may require additional angles in the requested interval.
This is why “press inverse sine” is only the beginning of solving an equation, not the end.
Solving Trigonometric Equations in an Interval
Suppose:
sin θ = 1/2, 0° ≤ θ ≤ 360°.
The reference angle is 30°. Sine is positive in Quadrants I and II, so:
θ = 30°, 150°.
The interval creates the final solution set. If the interval were larger, more coterminal solutions could appear.
A reliable equation route is:
- simplify to one trigonometric function if possible;
- find the reference or principal angle;
- determine all valid quadrants or use the graph;
- generate solutions;
- restrict to the stated interval;
- check in the original equation when transformations could introduce restrictions.
Worked Example 2: Solve a Cosine Equation
Solve 2 cos θ − 1 = 0 for 0° ≤ θ ≤ 360°.
cos θ = 1/2.
Reference angle = 60°. Cosine is positive in Quadrants I and IV:
θ = 60°, 300°.
If working in radians over 0 ≤ θ ≤ 2π, the corresponding answers are π/3 and 5π/3.
The Pythagorean Identities
The fundamental identity:
sin²θ + cos²θ = 1
comes directly from the unit circle. Divide every term by cos²θ:
tan²θ + 1 = sec²θ.
Divide instead by sin²θ:
1 + cot²θ = cosec²θ.
Remembering one identity plus the quotient/reciprocal relationships is often safer than memorising three unrelated lines.
Proving a Trigonometric Identity
An identity is a statement that is true for every value in its domain. A proof should therefore transform known relationships into the target relationship. Do not begin by assuming the identity is true and performing reversible-looking steps on both sides without control.
Useful proof strategies include:
- work on the more complicated side;
- convert sec, cosec, cot and tan into sine and cosine;
- use sin²θ + cos²θ = 1;
- factor or combine algebraic fractions;
- use conjugates where expressions contain 1 ± sin θ or 1 ± cos θ;
- stop once the target side has been reached.
A trigonometric identity proof is often an algebra problem wearing trigonometric notation.
Worked Example 3: Identity Proof
Show that:
(1 − cos²θ)/sin θ = sin θ.
Start from the left side. Use 1 − cos²θ = sin²θ:
(1 − cos²θ)/sin θ = sin²θ/sin θ = sin θ
for values where the original expression is defined. The proof is short because the correct identity was recognised immediately.
Compound-Angle Formulae
The syllabus uses the expansions:
sin(A+B) = sin A cos B + cos A sin B
sin(A−B) = sin A cos B − cos A sin B
cos(A+B) = cos A cos B − sin A sin B
cos(A−B) = cos A cos B + sin A sin B
For tangent:
tan(A+B) = (tan A + tan B)/(1 − tan A tan B)
tan(A−B) = (tan A − tan B)/(1 + tan A tan B).
Signs are a major failure point. Instead of memorising one vague pattern, derive the minus cases by replacing B with −B and using sin(−B)=−sin B, cos(−B)=cos B and tan(−B)=−tan B.
Worked Example 4: Exact Value Using a Compound Angle
Find the exact value of sin 75°.
Write 75° = 45° + 30°:
sin 75° = sin45°cos30° + cos45°sin30°
= (√2/2)(√3/2) + (√2/2)(1/2)
= (√6 + √2)/4.
The exact surd result shows how trigonometry reconnects with the surd chapter.
Double-Angle Formulae
Set B = A in the compound-angle formulae:
sin 2A = 2 sin A cos A
cos 2A = cos²A − sin²A
tan 2A = 2 tan A/(1 − tan²A).
Using sin²A + cos²A = 1, the cosine double-angle formula can also be written:
cos 2A = 2cos²A − 1 = 1 − 2sin²A.
Different forms are useful in different structures. If a question contains only cos²A, the 2cos²A−1 form may be natural. If it contains only sin²A, the 1−2sin²A form may reduce work.
R-Form: Compressing a Cosine-Sine Combination
An expression such as:
a cos θ + b sin θ
can be written as a single shifted sinusoid. Suppose:
R cos(θ − α).
Expand:
R cos θ cos α + R sin θ sin α.
Compare coefficients:
R cos α = a,
R sin α = b.
Squaring and adding gives:
R²(cos²α + sin²α) = a² + b²,
so R = √(a² + b²).
Also tan α = b/a, with quadrant/sign interpretation handled carefully according to the chosen form.
R-form is valuable because it reveals maximum and minimum possible values and simplifies certain equations.
Worked Example 5: R-Form and Maximum Value
Express 3 cos θ + 4 sin θ as R cos(θ − α).
First:
R = √(3² + 4²) = 5.
Compare:
5 cos α = 3,
5 sin α = 4.
So cos α = 3/5 and sin α = 4/5, giving tan α = 4/3. Therefore:
3 cos θ + 4 sin θ = 5 cos(θ − α), where α = tan⁻¹(4/3).
Since cosine lies between −1 and 1, the expression lies between −5 and 5. Its maximum possible value is therefore 5.
Worked Example 6: Solve Using an Identity
Solve 2sin²θ − 1 = 0 for 0° ≤ θ ≤ 360°.
2sin²θ = 1
sin²θ = 1/2
sin θ = ±√2/2.
The reference angle is 45°. Because both signs are allowed, sine can be positive or negative, producing all four quadrants:
θ = 45°, 135°, 225°, 315°.
A common error is to take a square root and keep only the positive branch. The ± must be preserved.
Trigonometric Modelling
Periodic phenomena can often be modelled with sine or cosine. The important modelling features are:
- midline: average level around which the system oscillates;
- amplitude: maximum deviation from the midline;
- period: length of one complete cycle;
- phase shift: horizontal positioning of the cycle;
- domain: the time or angle interval over which the model is intended.
Examples include idealised tides, seasonal quantities, rotating machinery and simple oscillatory motion. The student should not merely fit numbers into y = a sin(bx)+c. Each parameter should be interpreted.
Worked Example 7: Build a Simple Periodic Model
A quantity oscillates between 10 and 22 with period 12 units. A simple sine/cosine model needs:
Midline:
(22 + 10)/2 = 16.
Amplitude:
(22 − 10)/2 = 6.
If x is in radians and the period is 12, then b satisfies 2π/b = 12, so b = π/6. One possible model, depending on starting phase, is:
y = 16 + 6 sin(πx/6).
A different phase may be required if the initial condition is different. The model is constrained by both range and timing.
The Trigonometry Decision Tree
- Exact special angle? Use known exact values or derive them geometrically.
- Expression with sec/cosec/cot? Consider converting to sine and cosine.
- sin² and cos² both present? Consider a Pythagorean identity.
- Angle is A±B? Consider compound-angle formulae.
- Angle is 2A? Consider a double-angle form chosen to match the visible structure.
- a cos θ + b sin θ? Consider R-form.
- Equation? Simplify, find reference angle, generate all solutions, then apply the interval.
- Graph/model? identify amplitude, period, midline and phase before manipulating.
- Proof? transform one side using known identities; do not assume the target.
Common Failure Modes
| Visible error | Likely cause | Repair |
|---|---|---|
| Only one equation solution given | Calculator principal value treated as complete set | Use quadrants/graph and stated interval |
| sin⁻¹ confused with cosec | Inverse and reciprocal notation mixed | Write arcsin versus 1/sin explicitly |
| Wrong signs in compound formula | Formula memorised weakly | Use base formula and replace B by −B when needed |
| R-form coefficient mismatch | Expansion not compared term by term | Expand R cos(θ−α), then compare coefficients |
| Identity proof changes both sides together | Target assumed rather than proved | Work from one side to the other |
| Wrong graph period | bx and x/b transformations confused | Solve for one full repeat in the argument |
| Degrees/radians answer nonsense | Calculator mode mismatch | Write DEG or RAD before calculating |
Transfer Set
Question A
Find the exact value of cos 75°.
Answer: cos(45°+30°) = cos45°cos30° − sin45°sin30° = (√6−√2)/4.
Question B
Solve tan θ = −1 for 0° ≤ θ ≤ 360°.
Answer: reference angle 45°. Tangent is negative in Quadrants II and IV, so θ = 135°, 315°.
Question C
Simplify sec²θ − tan²θ.
Answer: from sec²θ = 1 + tan²θ, the result is 1.
Question D
Express 5 cos θ + 12 sin θ as R cos(θ−α) and state its maximum value.
Answer: R = √(25+144)=13. cos α = 5/13, sin α = 12/13. Maximum = 13.
Question E
For y = −4 cos(3x)+2 in degrees, state amplitude, period, maximum and minimum.
Answer: amplitude 4, period 120°, maximum 6, minimum −2.
A 50-Minute Trigonometry Repair Session
- 6 minutes: reconstruct exact values for 30°, 45°, 60° and convert them to radians.
- 6 minutes: retrieve reciprocal, quotient and Pythagorean identities.
- 8 minutes: solve four simple equations across a stated interval, forcing complete solution sets.
- 8 minutes: sketch two transformed trig graphs and label amplitude, period and midline.
- 8 minutes: complete one identity proof and one simplification.
- 7 minutes: calculate one compound-angle exact value and one double-angle expression.
- 5 minutes: convert one expression to R-form.
- 2 minutes: record the first error class: mode, sign, quadrant, identity, algebra or interval.
Trigonometry improves when the learner rotates among graph, unit-circle, algebra and function representations. Repeating only one surface creates brittle familiarity.
What Mastery Looks Like
- The learner moves between degrees and radians without calculator dependence.
- The learner knows exact special-angle values and can reconstruct them.
- The learner sees the six trig functions as a connected family.
- The learner reads transformed graphs through amplitude, period and midline.
- The learner distinguishes inverse functions from reciprocals.
- The learner gives all equation solutions in the required interval.
- The learner proves identities from known identities rather than circular algebra.
- The learner chooses compound, double-angle or R-form transformations based on visible structure.
- The learner can interpret a trig model rather than merely fit a formula.
Syllabus Alignment
This guide aligns with the 2027 Singapore-Cambridge SEC G3 Additional Mathematics syllabus section G1: six trigonometric functions for angles of any magnitude in degrees or radians; principal inverse-trig values; exact values for 30°, 45° and 60°; amplitude, periodicity and symmetry; prescribed sine, cosine and tangent graph forms; reciprocal, quotient and Pythagorean identities; compound-angle and double-angle formulae; R-form; simplification, equations in a given interval, simple identity proofs and trigonometric modelling.
Official SEAB 2027 G3 syllabus index
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