Secondary 3 Additional Mathematics | Singapore G3
Trigonometric Functions

Why Sine, Cosine and Tangent Keep Repeating Forever

A triangle angle ends at 90° or 180° depending on the problem. A trigonometric function does not. You can keep rotating forever.

This is the major Secondary 3 transition in trigonometry.

At lower levels, sine, cosine and tangent may feel like three buttons used inside right-angled triangles.

Additional Mathematics turns them into functions defined for angles of any magnitude. The angle can be positive, negative, larger than 360°, or measured in radians. Once trigonometry is viewed through rotation and the unit circle, periodicity stops being a rule to memorise and becomes inevitable.

rotate → return to the same direction → repeat the same trigonometric value.

The current Singapore G3 Additional Mathematics syllabus includes six trigonometric functions for angles of any magnitude in degrees or radians, principal values of inverse trigonometric functions, exact values for special angles, amplitude, periodicity and symmetry, transformed sine/cosine/tangent graphs, identities, equations and modelling. This page owns the function-and-graph foundation; the identities-and-equations machinery should be treated as the next dedicated teaching owner.

SEAB 2027 G3 Additional Mathematics syllabus (K341) →

The Topic Job

Turn trigonometry from a triangle-calculation tool into a family of periodic functions that describe rotation, oscillation and repeating change.

The Unit Circle Changes Everything

Take a circle of radius 1 centred at the origin.

A point on the circle at angle θ from the positive x-axis has coordinates:

(cos θ, sin θ).

This definition works far beyond acute angles.

Once the point moves around the circle:

• cos θ is the x-coordinate;
• sin θ is the y-coordinate;
• tan θ = sin θ / cos θ when cos θ ≠ 0.

The signs now come from coordinates, not from a separate mnemonic floating above the geometry.

The Six Trigonometric Functions

Function	Definition
sin θ	y-coordinate on unit circle
cos θ	x-coordinate on unit circle
tan θ	sin θ / cos θ
cosec θ	1 / sin θ
sec θ	1 / cos θ
cot θ	cos θ / sin θ = 1/tan θ

The reciprocal functions are not new geometry. They are built from sine, cosine and tangent.

Quadrants and Signs

On the unit circle:

Quadrant	sin	cos	tan
I	+	+	+
II	+	−	−
III	−	−	+
IV	−	+	−

Why is tangent positive in Quadrant III? Because both sin and cos are negative, so their ratio is positive.

A mnemonic can help retrieval, but the unit-circle signs should be able to reconstruct the answer.

Exact Values at Special Angles

The syllabus expects exact values for 30°, 45° and 60°—equivalently π/6, π/4 and π/3.

θ	sin θ	cos θ	tan θ
30°	1/2	√3/2	1/√3 = √3/3
45°	√2/2	√2/2	1
60°	√3/2	1/2	√3

These values can be rebuilt from 30°–60°–90° and 45°–45°–90° triangles rather than memorised as six unrelated fractions.

Degrees and Radians Measure the Same Angle

A full turn is:

360° = 2π radians.

Therefore:

180° = π radians.

Conversion:

• degrees → radians: multiply by π/180;
• radians → degrees: multiply by 180/π.

But radians are not merely another unit. They arise naturally from circle geometry.

What a Radian Means

For a circle of radius r and arc length s:

θ = s/r

when θ is measured in radians.

A one-radian angle subtends an arc whose length equals the radius.

This is why calculus formulas for sine and cosine work most cleanly in radians: the angle is directly tied to arc-length geometry.

Why Sine and Cosine Repeat

After one full rotation, a point on the unit circle returns to the same position.

Therefore:

sin(θ+2π)=sin θ
cos(θ+2π)=cos θ.

Sine and cosine have period 2π, or 360°.

Tangent repeats after half a turn because both sine and cosine change sign:

tan(θ+π)=tan θ.

So tangent has period π, or 180°.

The Sine Graph

For y=sin x:

• amplitude = 1;
• period = 2π;
• range = −1 ≤ y ≤ 1;
• passes through the origin;
• maximum 1 at π/2 + 2kπ;
• minimum −1 at 3π/2 + 2kπ.

The graph is a record of the unit-circle y-coordinate as the point rotates.

The Cosine Graph

For y=cos x:

• amplitude = 1;
• period = 2π;
• range = −1 ≤ y ≤ 1;
• starts at y=1 when x=0.

Cosine records the unit-circle x-coordinate.

The Tangent Graph

For y=tan x:

• period = π;
• range = all real numbers;
• undefined where cos x=0;
• vertical asymptotes occur at x=π/2+kπ.

The asymptotes are not arbitrary graph decorations. Tangent is sin/cos, so it becomes undefined where the denominator cos x is zero.

Amplitude

For:

y=a sin x

or:

y=a cos x,

the amplitude is |a|.

If a is negative, the graph is reflected as well as vertically scaled.

Changing the Period

For:

y=sin(bx)

or y=cos(bx), the period is:

2π/b.

Why? The function completes one full internal cycle when bx increases by 2π.

For y=tan(bx), the period is:

π/b.

Vertical Translation

For:

y=a sin(bx)+c,

the midline is y=c.

The range becomes:

c−|a| ≤ y ≤ c+|a|.

Worked Example 1 — Read a Transformed Sine Function

For:

y=3sin(2x)−4,

• amplitude = 3;
• period = 2π/2 = π;
• midline y=−4;
• maximum = −1;
• minimum = −7.

Do not sketch first and hope. Read the parameters first, then sketch using them.

Symmetry

Sine is an odd function:

sin(−x)=−sin x.

Cosine is an even function:

cos(−x)=cos x.

Tangent is odd:

tan(−x)=−tan x.

These symmetries are visible both on graphs and on the unit circle.

Inverse Trigonometric Functions

If:

sin θ = 1/2,

there are infinitely many angles with that sine value.

So sin⁻¹ cannot return every possible angle at once. The inverse function is defined to return a principal value.

Typical principal ranges are:

• sin⁻¹x: −π/2 ≤ θ ≤ π/2;
• cos⁻¹x: 0 ≤ θ ≤ π;
• tan⁻¹x: −π/2 < θ < π/2.

Your calculator’s inverse-trig button therefore gives one principal answer, not automatically every solution to a trigonometric equation.

sin⁻¹x Does Not Mean 1/sin x

This notation is a common trap.

• sin⁻¹x means inverse sine, arcsin x;
• 1/sin x is cosec x.

The same warning applies to cos⁻¹ and tan⁻¹.

Reference Angles

Suppose cos θ = −1/2.

The exact reference angle is 60°. Cosine is negative in Quadrants II and III.

So in 0°≤θ≤360°:

θ=120° or 240°.

This bridges the function foundation into the next canonical owner: trigonometric equations.

Modelling Repeating Change

Sine and cosine models can describe idealised repeating phenomena such as:

• height of a point on a rotating wheel;
• idealised tides over a limited interval;
• seasonal temperature patterns;
• alternating electrical signals;
• simple harmonic oscillation in later physics.

For a model:

h=A sin(bt)+c,

• |A| controls half the peak-to-trough range;
• 2π/b controls the period;
• c controls the midline.

A real phenomenon does not become a perfect sine wave just because it repeats. The model must be checked against data and context.

The Earliest Weak Link

What you see	Likely weak link
Correct acute-angle trig but wrong signs outside Quadrant I	triangle-only model not upgraded to unit circle
Confuses π/3 with 60π degrees	degree–radian conversion weak
Uses period 2π for tangent	does not derive period from function structure
Reads a in a sin bx+c as period	transformation parameters not separated
Treats sin⁻¹x as cosec x	inverse vs reciprocal notation confused
Calculator gives one angle and student stops	principal value mistaken for complete interval solution

Common Mistakes to Repair

• Calculator in the wrong angle mode. Degree/radian mode must match the question.
• Memorising quadrant signs without understanding coordinates.
• Using 2π/b for tangent. Tangent’s base period is π.
• Calling c the amplitude. c is the vertical translation / midline.
• Ignoring absolute value in amplitude. Amplitude is |a|.
• Confusing inverse and reciprocal notation.
• Assuming a periodic model repeats perfectly forever in reality.

Retrieval Check

1. Convert 150° to radians.
2. Convert 5π/6 to degrees.
3. State sin 30°, cos 45° and tan 60° exactly.
4. In which quadrants is cosine negative?
5. State the period of sin x and tan x.
6. For y=−4cos(3x)+2, state amplitude, period and midline.
7. Explain why tan x is undefined at π/2.

Transfer Set

1. Find the exact value of sin 150°.
2. Find the exact value of cos 225°.
3. For y=2sin(4x)−1, state amplitude, period, maximum and minimum.
4. Sketch one period of y=tan(2x), marking vertical asymptotes.
5. Given cos θ=−√3/2 and 0≤θ≤2π, find the possible θ values.
6. A rotating point has height h=5+3sin(πt/4). State the midline, amplitude and period.

Answer outline — open only after attempting

1. 1/2.
2. −√2/2.
3. Amplitude 2; period π/2; maximum 1; minimum −3.
4. Period π/2; asymptotes where 2x=π/2+kπ, so x=π/4+kπ/2.
5. θ=5π/6, 7π/6.
6. Midline 5; amplitude 3; period 2π/(π/4)=8.

Independent Checks

• Check degree/radian mode.
• Use the unit circle to verify the sign.
• For exact values, estimate the decimal only as a reasonableness check.
• For transformed graphs, verify range from c±|a|.
• Verify period by testing whether the function value repeats after the proposed interval.
• For a model, check units and whether the period makes contextual sense.

For Parents and Tutors — What This Topic Is Really Testing

The critical transition is from triangle trigonometry to function trigonometry.

If a student can only use SOH-CAH-TOA inside a drawn right triangle, later identities and equations will feel like a different subject.

Ask the student to explain:

• why sin is the unit-circle y-coordinate;
• why cosine is negative in Quadrant II;
• why sine repeats after 2π;
• why tangent repeats after π;
• why tan has vertical asymptotes;
• why inverse trig returns a principal value rather than every possible angle.

The teaching target is rotation → coordinate → function → graph → repeat.

Once that model is stable, trigonometric identities and equations have somewhere to attach.

Where This Connects Next

• Secondary 3 Additional Mathematics: Complete A-Math Map
• Additional Mathematics Directory
• Surds | Exact Irrational Numbers
• Trigonometric Identities & Equations

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Bukit Timah Tutor Mathematics
The triangle is only the beginning. Once an angle becomes rotation, trigonometry becomes a function that can travel around the circle forever.