Why does sin(3x) complete a cycle sooner, while sin(x/3) takes longer? In both cases, the sine function repeats when its input advances by 2π radians. For sin(3x), x needs to increase by only 2π/3. For sin(x/3), x must increase by 6π. The graph changes because the angle entering the function changes at a different rate.
This is the central habit of the chapter: read what happens to the input before deciding what happens to the graph. A coefficient outside the function changes the output. A coefficient inside changes how quickly the input travels through a cycle. A constant added afterwards changes the vertical position.
This guide connects exact values, radians, the six trigonometric functions, principal inverse values and graph transformations. It includes worked sketches described through key points, diagnostic counterexamples and original practice with complete solutions. You do not need to memorise an unrelated picture for every formula.
Curriculum scope: trigonometric functions and graphs appear in the 2026 O-Level 4049, 2026 N(A)-Level 4051, 2027 SEC G3 K341 and 2027 SEC G2 K232 syllabuses. The listed graph families include scaled and vertically translated sine and cosine functions and scaled tangent functions. The short phase-shift section below is labelled as an extension, not another claim about the explicitly listed graph families.
Navigate: functions and exact values · radians · inverse values · transformations · worked graphs · mistake clinic · practice · solutions.
Start with one point moving around a unit circle
Imagine a circle of radius one centred at the origin. A radius makes an angle θ with the positive horizontal axis, measured anticlockwise for a positive angle. The endpoint has coordinates (cos θ, sin θ). Cosine records its horizontal coordinate; sine records its vertical coordinate.
This description extends beyond the acute angles of a right-angled triangle. The point can move through every quadrant, return after a complete turn, or move clockwise through a negative angle. The right-triangle ratios remain useful, but the circle explains signs, repetition and angles beyond 90°.
For the six functions, use these relationships wherever the denominators are nonzero:
| Function | Relationship | When it is undefined |
|---|---|---|
| sin θ | Vertical unit-circle coordinate | Defined for every real angle |
| cos θ | Horizontal unit-circle coordinate | Defined for every real angle |
| tan θ | sin θ / cos θ | When cos θ = 0 |
| cosec θ | 1 / sin θ | When sin θ = 0 |
| sec θ | 1 / cos θ | When cos θ = 0 |
| cot θ | cos θ / sin θ | When sin θ = 0 |
For example, sec 60° = 1/(1/2) = 2, while sec 90° is undefined because cos 90° = 0. “Undefined” is not another name for zero. It says that the defining division cannot produce a real value at that input.
Some textbooks write csc instead of cosec. Both notations refer to the reciprocal of sine. Keep the meaning stable even when the notation changes.
Exact values from two familiar triangles
A right-angled isosceles triangle with legs 1 and 1 has hypotenuse √2. Its acute angles are 45°, so sin 45° = cos 45° = 1/√2 = √2/2 and tan 45° = 1.
Next, split an equilateral triangle of side 2 into two congruent right-angled triangles. Each has hypotenuse 2, short leg 1 and long leg √3. This gives sin 30° = 1/2, cos 30° = √3/2, sin 60° = √3/2 and cos 60° = 1/2. Dividing sine by cosine gives the tangent values.
| Angle | Sine | Cosine | Tangent |
|---|---|---|---|
| 0° | 0 | 1 | 0 |
| 30° | 1/2 | √3/2 | √3/3 |
| 45° | √2/2 | √2/2 | 1 |
| 60° | √3/2 | 1/2 | √3 |
| 90° | 1 | 0 | Undefined |
Use these as exact quantities rather than replacing every surd with a decimal. A decimal approximation can help check size, but √3/2 carries a precise relationship that a rounded 0.866 does not preserve exactly.
Quadrants tell you the signs
In the first quadrant, both coordinates are positive. In the second, the horizontal coordinate is negative and the vertical coordinate is positive. In the third, both are negative. In the fourth, the horizontal coordinate is positive and the vertical coordinate is negative.
Therefore sine is positive in the first and second quadrants, cosine in the first and fourth, and tangent in the first and third. Reciprocal functions have the same signs as the functions they reciprocate, where defined.
For sin 150°, the reference angle is 30° and the point is in the second quadrant, so sin 150° = 1/2. For cos 225°, the reference angle is 45° in the third quadrant, giving −√2/2. For tan 330°, the reference angle is 30° in the fourth quadrant, giving −√3/3.
The reference angle supplies the magnitude; the quadrant supplies the sign. Using a calculator without understanding that division can hide an error until the same idea appears in a graph or an exact-value question.
Radians measure an angle through arc and radius
An angle of one radian subtends an arc whose length equals the circle’s radius. More generally, for a circular arc of length s and radius r, the angle in radians is θ = s/r. Because a complete circumference is 2πr, one complete turn is 2π radians.
Thus 180° = π radians. To convert degrees to radians, multiply by π/180. To convert radians to degrees, multiply by 180/π.
For example, 150° = 150π/180 = 5π/6. Conversely, 7π/4 radians = (7π/4)(180/π) = 315°. The angle has not changed; only its unit has.
On a radian graph, label important inputs as π/2, π, 3π/2 and 2π when appropriate. On a degree graph, use 90°, 180°, 270° and 360°. Do not combine a radian formula with degree calculator mode.
A quick mode check is sin(π/6) in radian mode or sin 30° in degree mode: both should give 1/2. The expression sin 30 without a unit is ambiguous in informal work, so make the convention explicit at the beginning of the question.
Symmetry explains more than a remembered sign rule
Reflecting a unit-circle point across the horizontal axis changes θ to −θ. Its horizontal coordinate stays the same and its vertical coordinate changes sign. Therefore cos(−θ) = cos θ and sin(−θ) = −sin θ.
Cosine is an even function: its graph is symmetric about the vertical axis. Sine is an odd function: its graph has rotational symmetry through the origin. Tangent is also odd because it is sine divided by cosine wherever defined.
After a full turn, the sine and cosine coordinates repeat. After a half-turn, both coordinates change sign, so their ratio stays unchanged. This explains why the basic tangent function repeats after π rather than 2π.
These are checks as well as facts to learn. A sketch of y = cos x that is not symmetric about the vertical axis has a problem. A sketch of y = sin x that begins at height 1 when x = 0 has confused sine with cosine.
An inverse function is not a reciprocal
The notation sin−1 u usually means arcsin u, the principal inverse sine. It does not mean 1/sin u. The reciprocal of sine is cosec. This distinction affects calculator keys, written working and the interpretation of answers.
Sine repeats, so many angles can have the same sine. To make inverse sine return one value, we restrict its output to a principal interval. The corresponding principal intervals are:
| Inverse | Allowed input | Principal output in radians |
|---|---|---|
| arcsin u | −1 ≤ u ≤ 1 | −π/2 to π/2, inclusive |
| arccos u | −1 ≤ u ≤ 1 | 0 to π, inclusive |
| arctan u | Any real u | Strictly between −π/2 and π/2 |
Consequently, arcsin(sin 150°) = 30°, not 150°, because inverse sine returns its principal value. Similarly, arccos(cos 240°) = 120° and arctan(tan 135°) = −45°.
An inverse calculator value is therefore not automatically the complete answer to a trigonometric equation on a specified interval. This guide establishes the function and graph meaning. For a full interval-solving method, use the separate Trigonometric Equations guide.
Read a transformed sine or cosine in three passes
For y = a sin(bx) + c with nonzero a and positive b, first read the vertical centre c. Next read the amplitude |a|, which is the distance from the centre to a maximum or minimum. Finally, determine how much x must change for bx to increase by 2π.
The period is therefore 2π/b radians, or 360°/b when x is measured in degrees. The complete range is c − |a| ≤ y ≤ c + |a|. Cosine follows the same amplitude, centre and period rules.
For y = a sin(x/b) + c, the input changes more slowly. A full sine cycle requires x/b to increase by 2π, so x increases by 2πb. The amplitude and vertical centre have not changed.
| Function | Vertical behaviour | Period in radians |
|---|---|---|
| a sin(bx) + c | Centre c; amplitude |a| | 2π/b |
| a cos(bx) + c | Centre c; amplitude |a| | 2π/b |
| a sin(x/b) + c | Centre c; amplitude |a| | 2πb |
| a cos(x/b) + c | Centre c; amplitude |a| | 2πb |
| a tan(bx) | Unbounded; no amplitude | π/b |
The period formulas in this table assume a ≠ 0 and b > 0. The named school graph families use positive integer b. When a = 0 in a sine or cosine expression, the result is the constant function y = c. Its amplitude is zero, but it has no least positive period: every positive shift leaves it unchanged.
A negative a reflects the output about the horizontal axis before the vertical translation is added. In y = −3 cos x + 2, the graph begins at −1 rather than 5 because cos 0 = 1. Amplitude is a nonnegative distance, so it is 3, not −3.
Period, frequency and range are different questions
The period measures the input interval needed for repetition. When the input represents time, frequency in cycles per unit time is the reciprocal of the period. Amplitude measures output displacement from the centre. Range records the possible output values.
A restricted input interval may not contain the graph’s full range. For instance, y = sin x on 0 ≤ x ≤ π/6 has range 0 ≤ y ≤ 1/2, even though the unrestricted sine function ranges from −1 to 1. Read the stated domain before reporting maxima and minima.
Worked graph 1: y = 2 sin(3x) − 1
Sketch one cycle for 0 ≤ x ≤ 2π/3. The centre is −1, the amplitude is 2, and the period is 2π/3. The complete range is −3 ≤ y ≤ 1.
A sine cycle is easy to organise through input angles 0, π/2, π, 3π/2 and 2π. Here the input angle is 3x, so divide each of those angles by 3 to obtain the x-coordinates.
| x | 3x | y |
|---|---|---|
| 0 | 0 | −1 |
| π/6 | π/2 | 1 |
| π/3 | π | −1 |
| π/2 | 3π/2 | −3 |
| 2π/3 | 2π | −1 |
Mark the centre line y = −1, label the maximum and minimum, and join the points with the smooth sine shape. Do not connect them as a zigzag made of straight segments. The key points locate the curve; they do not replace its continuous shape.
The graph crosses the horizontal axis when sin(3x) = 1/2. Within this single cycle, the input angles are π/6 and 5π/6, giving x = π/18 and 5π/18. These crossings lie between the key points, which is why a five-point table alone may not list every feature requested in a sketch.
Check the first point directly: y(0) = −1. A sketch beginning at zero has forgotten the vertical translation. Check the maximum: 2(1) − 1 = 1. A maximum labelled 2 has confused amplitude with maximum height.
Worked graph 2: y = −3 cos(x/2) + 2
Sketch one cycle for 0 ≤ x ≤ 4π. The centre is 2, amplitude 3 and period 4π. Its range is −1 ≤ y ≤ 5.
Set x/2 equal to the standard quarter-cycle cosine angles. This gives x = 0, π, 2π, 3π and 4π. The corresponding outputs are −1, 2, 5, 2 and −1.
The negative coefficient reverses the usual cosine starting height. At x = 0, cosine is 1, so y = −3 + 2 = −1, a minimum. At x = 2π, cosine receives π and equals −1, so y = 3 + 2 = 5, a maximum.
Compare this with −3 cos(2x) + 2. Its vertical features are identical, but its period would be π, not 4π. The difference is inside the brackets. Reading x/2 as though it were 2x changes the cycle length by a factor of four.
Worked graph 3: a degree-based question
Sketch y = 2 cos(2x) + 1 on 0° ≤ x ≤ 180°. Here x is measured in degrees. A complete cosine cycle occurs when 2x advances through 360°, so the period is 180°.
The centre is 1, the amplitude is 2 and the range is −1 ≤ y ≤ 3. At x = 0°, 45°, 90°, 135° and 180°, the values are 3, 1, −1, 1 and 3.
The arithmetic resembles the radian examples, but the axis labels and calculator mode differ. You may convert the whole question into radians if appropriate, but you must then convert all relevant inputs consistently. A graph cannot use 90 and π/2 as though they were different positions for the same angle convention.
Worked graph 4: tangent has branches, not a wave amplitude
Consider y = 2 tan(2x). Its period is π/2. The central branch lies between x = −π/4 and x = π/4 because tangent becomes undefined when its input reaches −π/2 or π/2.
Three useful points are (−π/8, −2), (0,0) and (π/8,2). As x approaches π/4 from the left, the output increases without bound. As x approaches −π/4 from the right, it decreases without bound.
Draw the two vertical asymptotes as boundaries the curve approaches, not as part of the function. Draw a smooth increasing branch through the three points. Do not join it across an asymptote to the next branch.
Tangent has no finite maximum or minimum over a complete branch, so it has no amplitude. The outside factor 2 stretches the output values; it does not create a highest point of 2. On a larger interval, locate every relevant asymptote and repeat the appropriate branches.
Worked graph 5: reconstruct a function from its features
A sinusoidal graph has maximum 7 and minimum −1. Consecutive maxima occur at x = 0 and x = π. Give a cosine model.
The centre is the average of the extreme values: c = (7 + (−1))/2 = 3. The amplitude is half their difference: |a| = (7 − (−1))/2 = 4.
The period is π, so 2π/b = π and b = 2. Because a maximum occurs at x = 0, choose a positive cosine coefficient. One suitable model is y = 4 cos(2x) + 3.
Substitute x = 0 to obtain 7 and x = π/2 to obtain −1. These checks connect the equation back to the features that determined it. Equivalent expressions may exist; the question here asks for one appropriate cosine model.
A sampling trap: five zeros do not prove a straight line
Suppose someone samples y = sin(2x) at x = 0, π/2, π, 3π/2 and 2π. Every sampled value is zero. Joining those points along the horizontal axis would erase the oscillation completely.
The mistake is not a wrong sine value. It is a poor choice of sample positions. Between 0 and π/2, the curve reaches 1 at x = π/4. Between π/2 and π, it reaches −1 at x = 3π/4.
Use the function’s period to choose quarter-cycle points. A table selected without reference to the input transformation can hide precisely the behaviour a sketch is supposed to show.
A labelled extension: what a horizontal shift means
The explicitly listed elementary graph families do not require us to treat every phase-shift form as another core memorisation rule. Still, one controlled extension helps explain later connections, including R-form.
Compare y = 2 sin(3x) − 1 with y = 2 sin(3(x − π/6)) − 1. The second uses x − π/6 where the first used x. Its corresponding points occur π/6 farther to the right. Its amplitude, centre and period remain 2, −1 and 2π/3.
Expanding the inner bracket gives 3x − π/2. The horizontal shift is π/6, not π/2, because the coefficient 3 also acts on the shifted input. Factoring the inner expression reveals the actual shift.
Keep this as a bridge rather than an excuse to overload the chapter before the core graphs are secure. The separate R-Form guide owns the fuller combination-of-sine-and-cosine task.
A hypothetical periodic model
Suppose a simplified height model is h = 4 + 2 cos(πt/3) for 0 ≤ t ≤ 6, where t is measured in hours and h in metres. This is an invented mathematical model, not an observation of a particular tide or machine.
The mean level is 4 metres and the amplitude is 2 metres. The angle πt/3 advances by 2π when t increases by 6, so the period is 6 hours. Heights at t = 0, 1.5, 3, 4.5 and 6 are 6, 4, 2, 4 and 6 metres.
The coefficient π/3 has the role of radians per hour in this interpretation. It is not an amplitude. The model’s minimum is 2 metres, not −2, because the oscillation occurs around the level 4.
A real periodic process may not have a perfectly constant amplitude, period or mean. The simple equation makes a pattern inspectable, but its usefulness depends on whether those assumptions fit the situation. Do not turn this illustrative curve into an unsupported claim about the real world.
Mistake clinic: identify which feature went wrong
“The amplitude of −3 cos x + 2 is −3.” Amplitude is a distance and equals 3. The minus sign describes reflection. Repair the distinction between size and orientation.
“The maximum is the amplitude.” Only when the centre is zero. For 2 sin(3x) − 1, the amplitude is 2 but the maximum is 1. Calculate centre plus amplitude and centre minus amplitude.
“The period of sin(3x) is 6π.” That is the period of sin(x/3). Solve 3 × change in x = 2π to locate the correct period. This input-based explanation is safer than a half-remembered rule.
“The inverse key gives every solution.” The inverse gives a principal value. The graph and stated interval determine whether other angles are relevant. Do not erase the principal-value convention or confuse it with a reciprocal.
“Undefined means zero.” tan 90° is undefined, while tan 0° is zero. In the first case its cosine denominator vanishes; in the second the sine numerator vanishes and the denominator does not.
“Tangent’s outside coefficient is its amplitude.” Tangent is unbounded on a full branch. The coefficient changes scale, not a maximum height. Repair by checking values close to an asymptote rather than trying to add a sine-wave label.
“The full function range is always the answer.” A restricted domain may cover only part of a cycle. Evaluate its endpoints and relevant turning points before reporting the range on that interval.
Independent practice
State whether each angle is in degrees or radians. For graph questions, identify the centre, amplitude where applicable, period, domain restrictions and a useful set of key points before drawing.
1. Convert 150° to radians and 7π/4 radians to degrees.
2. Find the exact values of sin 150°, cos 225° and tan 330°.
3. Find sec 60°, cosec 30° and cot 45°. Explain why sec 90° is undefined.
4. State the amplitude, centre, period and full range of y = 3 sin(2x) + 4, with x in radians.
5. State the period and full range of y = 2 cos(x/3) − 1.
6. Find y = −4 sin x + 2 at x = 0, π/2, π, 3π/2 and 2π. Use the results to describe one cycle.
7. Find the period and the two asymptotes bounding the central branch of y = tan(3x).
8. A sinusoidal graph has maximum 5, minimum −3 and consecutive maxima at x = 0 and x = 2π/3. Give a cosine model.
9. Find the principal values, in radians, of arcsin(−1/2), arccos(−1/2) and arctan(−1).
10. Find the horizontal-axis crossings of y = 2 sin x + 1 on 0 ≤ x ≤ 2π.
11. Explain why y = 4 + cos(2x) has no horizontal-axis crossings. State its period.
12. Compare y = 2 cos x and y = 2 cos(2x). Which features are unchanged, and which change?
13. A hypothetical model is h = 5 − 3 cos(πt/2), for 0 ≤ t ≤ 4. Find its period, minimum, maximum and value at t = 1.
14. A student substitutes a = 0 into y = a sin(2x) + 3 and reports that its least positive period is π. Explain the problem.
Full solutions
1–3: units, exact values and reciprocal functions
1. Multiply 150 by π/180 to obtain 5π/6. Multiply 7π/4 by 180/π to obtain 315°. These conversions change the unit, not the angle.
2. The reference angles are 30°, 45° and 30°. The relevant quadrant signs give 1/2, −√2/2 and −√3/3 respectively.
3. sec 60° = 1/cos 60° = 2; cosec 30° = 1/sin 30° = 2; cot 45° = cos 45°/sin 45° = 1. sec 90° would require division by cos 90° = 0, so it is undefined.
4–7: read the transformation
4. Amplitude 3, centre 4, period π, and full range 1 ≤ y ≤ 7. The input 2x completes one sine cycle as x increases by π.
5. The argument is x/3, so the period is 6π. The centre is −1 and amplitude 2, giving the full range −3 ≤ y ≤ 1.
6. The values are 2, −2, 2, 6 and 2. The graph starts on its centre line, moves downward to its minimum at π/2, returns to the centre at π, reaches its maximum at 3π/2 and returns to the centre at 2π. The reflection reverses the usual initial sine direction.
7. Tangent repeats when 3x advances by π, so the period is π/3. The central branch is bounded by x = −π/6 and x = π/6, obtained from 3x = −π/2 and π/2.
8–10: reconstruct and interpret
8. The centre is (5 + (−3))/2 = 1 and the amplitude is (5 − (−3))/2 = 4. The period 2π/3 gives b = 3. Starting at a maximum, one model is y = 4 cos(3x) + 1.
9. The principal outputs are −π/6, 2π/3 and −π/4. The arccos answer is positive because its principal interval is from 0 to π.
10. Set 2 sin x + 1 = 0, giving sin x = −1/2. Within the stated interval, sine is negative in the third and fourth quadrants. The crossings are x = 7π/6 and x = 11π/6. Neither endpoint is a crossing because the output there is 1.
11–14: check features and special cases
11. Since −1 ≤ cos(2x) ≤ 1, the output satisfies 3 ≤ y ≤ 5. Zero is not in this range, so there are no horizontal-axis crossings. The period is π.
12. Both have centre zero, amplitude 2, full range −2 to 2 and a maximum at x = 0. Their periods differ: 2π for 2 cos x and π for 2 cos(2x). The second completes twice as many cycles over the same input interval.
13. The angle πt/2 advances by 2π when t increases by 4. The minimum height is 2 at t = 0 and t = 4; the maximum is 8 at t = 2. At t = 1, cosine receives π/2, so h = 5.
14. With a = 0, the expression is simply y = 3. Every positive input shift preserves it, so there is no least positive period. The nonconstant sine period rule cannot be used to claim a fundamental period for this constant function.
How to turn a wrong graph into a useful diagnosis
Separate the features instead of labelling the entire sketch wrong. A correct period with wrong heights suggests a vertical-transformation error. Correct heights with too many cycles suggests an input-scaling error. Correct key points joined across a tangent asymptote suggests a continuity or domain error.
Ask the student to identify one point that must be correct, such as x = 0, and one feature that must be true, such as the range. Use these as anchors for reconstruction. A graph should be checked against its equation, not merely compared with a remembered picture.
A short retest can change only one part: replace sin(3x) by sin(x/3), change a positive outside coefficient to a negative one, or move the centre from zero to two. The student should predict the changed feature before plotting any values.
After those controlled changes are secure, mix sine, cosine and tangent without announcing which rule to use. The student must identify the function family, read the input, read the output transformation and respect its domain. This is the step from copying a sketch to constructing one.
What comes next?
Graphs, identities and equations are connected, but they are not the same task. A graph describes a relationship over an input range. An identity establishes an equality wherever both sides are defined. An equation asks which inputs satisfy a specified condition.
Use the Trigonometric Identities guide for justified transformations, and the Trigonometric Equations guide for interval solutions. The Geometry and Trigonometry strand guide connects those jobs within the wider subject.
Return to the Additional Mathematics Hub when the learning question changes. Families considering help can use the separate Additional Mathematics Tuition page. A graph with the student’s own axis labels and working is more informative than a score alone.
Source and scope review: 6 September 2026. The linked official syllabuses establish the applicable subject-content boundaries. All worked examples, graph-point tables, practice questions and hypothetical models here are original teaching material. The aim is dependable interpretation and independent reconstruction, not a promise of a particular examination result.
Continue the trigonometry route across eduKate
For another worked teaching route, continue to the eduKateSengkang Trigonometric Functions, Identities, Equations and Graphs guide.
When the difficulty spans identities, equations, graph behaviour and geometric interpretation, use the Bukit Timah Tutor Trigonometric Functions, Identities, Equations and Geometry Synthesis Guide.