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Trigonometric graphs and models

Read input/output scaling, preserve tangent discontinuities and infer a bounded periodic model with correct phase units.

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Pause after each idea, then explain it in your own words before moving on.

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Enlarged diagram

Scaling input and output have different effects

In y = a sin(bx) + c, with a ≠ 0 and b > 0, the sine value is multiplied by a and then shifted by c. Sine still runs from −1 to 1, so the wave has midline y = c, amplitude |a| and range c − |a| to c + |a|. A negative a reflects the wave in its midline; it does not give a negative amplitude.

The input coefficient changes how quickly the angle completes a turn. For b > 0, sin(bx) completes one cycle when bx increases by 2π radians, so x increases by 2π/b. In sin(x/b), x must instead increase by 2πb to produce that same full turn.

Form, with a ≠ 0 and b > 0Period when x is in radiansPeriod when x is in degrees
a sin(bx) + c or a cos(bx) + c2π/b360°/b
a sin(x/b) + c or a cos(x/b) + c2πb360°b

Cosine follows the same amplitude and period rules, but has a different starting value. A sine wave starts at its midline at x = 0; a cosine wave starts at c + a. Thus a negative cosine coefficient makes it start at its minimum.

Halving the input rate doubles the period

Two sine waves share their vertical range and start at output 1. The solid wave reaches its first maximum at x = π/2 and repeats after 2π; the dashed wave reaches its first maximum at π and repeats after 4π.

Scroll across the figure to read all labels.

Two sine waves share their vertical range and start at output 1. The solid wave reaches its first maximum at x = π/2 and repeats after 2π; the dashed wave reaches its first maximum at π and repeats after 4π.Two sine waves share their vertical range and start at output 1. The solid wave reaches its first maximum at x = π/2 and repeats after 2π; the dashed wave reaches its first maximum at π and repeats after 4π.
The solid curve y = 2 sin x + 1 makes two complete cycles from 0 to 4π radians. The dashed curve y = 2 sin(x/2) + 1 makes one. Both have midline 1, amplitude 2 and range −1 to 3; their periods are 2π and 4π, respectively.
Open full-size graphOpen full-size graph
What to noticeThe x-axis and table use decimal radians: approximately 1.571 is π/2, 3.142 is π and 6.283 is 2π. Read a period between points with the same height and direction of motion, not just any two midline crossings. The exact quarter-cycle inputs are tabulated in the lesson.
View figure data
Figure data: Halving the input rate doubles the period
Angle x (rad)y = 2 sin x + 1y = 2 sin(x/2) + 1
0.00001.00001.0000
0.13091.26111.1308
0.26181.51761.2611
0.39271.76541.3902
0.52362.00001.5176
0.65452.21751.6429
0.78542.41421.7654
0.91632.58671.8846
1.04722.73212.0000
1.17812.84782.1111
1.30902.93192.2175
1.43992.98292.3187
1.57083.00002.4142
1.70172.98292.5037
1.83262.93192.5867
1.96352.84782.6629
2.09442.73212.7321
2.22532.58672.7937
2.35622.41422.8478
2.48712.21752.8939
2.61802.00002.9319
2.74891.76542.9616
2.87981.51762.9829
3.01071.26112.9957
3.14161.00003.0000
3.27250.73892.9957
3.40340.48242.9829
3.53430.23462.9616
3.66520.00002.9319
3.7961-0.21752.8939
3.9270-0.41422.8478
4.0579-0.58672.7937
4.1888-0.73212.7321
4.3197-0.84782.6629
4.4506-0.93192.5867
4.5815-0.98292.5037
4.7124-1.00002.4142
4.8433-0.98292.3187
4.9742-0.93192.2175
5.1051-0.84782.1111
5.2360-0.73212.0000
5.3669-0.58671.8846
5.4978-0.41421.7654
5.6287-0.21751.6429
5.75960.00001.5176
5.89050.23461.3902
6.02140.48241.2611
6.15230.73891.1308
6.28321.00001.0000
6.41411.26110.8692
6.54501.51760.7389
6.67591.76540.6098
6.80682.00000.4824
6.93772.21750.3571
7.06862.41420.2346
7.19952.58670.1154
7.33042.73210.0000
7.46132.8478-0.1111
7.59222.9319-0.2175
7.72312.9829-0.3187
7.85403.0000-0.4142
7.98492.9829-0.5037
8.11582.9319-0.5867
8.24672.8478-0.6629
8.37762.7321-0.7321
8.50852.5867-0.7937
8.63942.4142-0.8478
8.77032.2175-0.8939
8.90122.0000-0.9319
9.03211.7654-0.9616
9.16301.5176-0.9829
9.29391.2611-0.9957
9.42481.0000-1.0000
9.55570.7389-0.9957
9.68660.4824-0.9829
9.81750.2346-0.9616
9.94840.0000-0.9319
10.0793-0.2175-0.8939
10.2102-0.4142-0.8478
10.3411-0.5867-0.7937
10.4720-0.7321-0.7321
10.6029-0.8478-0.6629
10.7338-0.9319-0.5867
10.8647-0.9829-0.5037
10.9956-1.0000-0.4142
11.1265-0.9829-0.3187
11.2574-0.9319-0.2175
11.3883-0.8478-0.1111
11.5192-0.73210.0000
11.6501-0.58670.1154
11.7810-0.41420.2346
11.9119-0.21750.3571
12.04280.00000.4824
12.17370.23460.6098
12.30460.48240.7389
12.43550.73890.8692
12.56641.00001.0000

For y = 2 sin x + 1, the range is [−1,3] and the period 2π. The wave y = 2 sin(x/2) + 1 has the same range but takes twice as much x to complete each cycle. These quarter-cycle points show the mapping exactly:

Phase inside sinex for 2 sin x + 1x for 2 sin(x/2) + 1Output on either wave
0001
π/2π/2π3
ππ2π1
3π/23π/23π−1
2π2π4π1

A period returns to the same position and direction in the cycle. The two midline crossings within a sine cycle have opposite directions, so their separation is only half a period. Increasing output by c changes none of these input spacings.

The unit-circle symmetries also appear in graphs: sin(−x) = −sin x makes the unshifted sine graph symmetric about the origin, while cos(−x) = cos x makes cosine symmetric about the vertical axis. Shifting a sine wave up by c moves its centre of rotational symmetry to (0,c).

Check your understanding 1 The bracket determines the period

Give the amplitude, range and period in degrees of y = -3 cos(2x) + 4. Compare its period with y = -3 cos(x/2) + 4. Explain the starting value and why the amplitude is positive in both cases.

Show answer

The amplitude is 3 and range [1,7]. The first period is 180° and the second 720°. Both start at y = 1, their minimum, because cosine is 1 at input zero and its coefficient is −3. The negative coefficient reflects the wave; its amplitude is the positive distance 3 from the midline y = 4. The input rates, rather than vertical scaling, distinguish the periods.

Tangent has a different repeating pattern

For y = a tan(bx) with a ≠ 0 and b > 0, the period is π/b radians, or 180°/b. Tangent repeats after a half-turn of its input because both sine and cosine change sign, leaving their ratio unchanged. It has no finite amplitude or greatest value: the ratio is unbounded near inputs where cosine is zero.

For y = tan(2x), the branch containing the origin lies between x = −π/4 and π/4. Those endpoints make cos(2x) = 0, so they are excluded. The branch crosses (0,0), passes through (−π/8,−1) and (π/8,1), and increases without bound near the right asymptote. The neighbouring branch is separate even though its shape repeats.

Tangent branches stay separate across excluded inputs

Three increasing tangent branches are drawn separately over x from −π/2 to π/2 radians. The central branch rises between the two excluded inputs −π/4 and π/4. Adjacent branches lie on the other sides of those asymptotes, with no connecting function stroke across either gap.

Scroll across the figure to read all labels.

Three increasing tangent branches are drawn separately over x from −π/2 to π/2 radians. The central branch rises between the two excluded inputs −π/4 and π/4. Adjacent branches lie on the other sides of those asymptotes, with no connecting function stroke across either gap.Three increasing tangent branches are drawn separately over x from −π/2 to π/2 radians. The central branch rises between the two excluded inputs −π/4 and π/4. Adjacent branches lie on the other sides of those asymptotes, with no connecting function stroke across either gap.
The three curve portions all belong to y = tan(2x). The central branch crosses (0,0) and passes through (−π/8,−1) and (π/8,1). The vertical dotted lines at x = −π/4 and π/4 are asymptotes, not parts of the function. The repeating horizontal interval is π/2 radians.
Open full-size graphOpen full-size graph
What to noticeThe visible output window is −4 to 4. Curve tips at the edge of that window are cropped portions of unbounded branches, not endpoints or extrema. Different line styles distinguish the three portions of the same function. The two labelled vertical lines mark excluded inputs; their table rows describe those guide lines, not defined function values.
View figure data
Figure data: Tangent branches stay separate across excluded inputs
SeriesAngle x (rad)Output y (dimensionless)
Left branch: y = tan(2x)-1.570800.00000
Left branch: y = tan(2x)-1.550080.04146
Left branch: y = tan(2x)-1.529360.08305
Left branch: y = tan(2x)-1.508650.12494
Left branch: y = tan(2x)-1.487930.16726
Left branch: y = tan(2x)-1.467220.21017
Left branch: y = tan(2x)-1.446500.25384
Left branch: y = tan(2x)-1.425790.29844
Left branch: y = tan(2x)-1.405070.34415
Left branch: y = tan(2x)-1.384350.39119
Left branch: y = tan(2x)-1.363640.43977
Left branch: y = tan(2x)-1.342920.49017
Left branch: y = tan(2x)-1.322210.54265
Left branch: y = tan(2x)-1.301490.59755
Left branch: y = tan(2x)-1.280770.65523
Left branch: y = tan(2x)-1.260060.71614
Left branch: y = tan(2x)-1.239340.78078
Left branch: y = tan(2x)-1.218630.84974
Left branch: y = tan(2x)-1.197910.92373
Left branch: y = tan(2x)-1.177191.00362
Left branch: y = tan(2x)-1.156481.09044
Left branch: y = tan(2x)-1.135761.18549
Left branch: y = tan(2x)-1.115051.29036
Left branch: y = tan(2x)-1.094331.40708
Left branch: y = tan(2x)-1.073611.53827
Left branch: y = tan(2x)-1.052901.68732
Left branch: y = tan(2x)-1.032181.85880
Left branch: y = tan(2x)-1.011472.05891
Left branch: y = tan(2x)-0.990752.29637
Left branch: y = tan(2x)-0.970042.58379
Left branch: y = tan(2x)-0.949322.94018
Left branch: y = tan(2x)-0.928603.39550
Left branch: y = tan(2x)-0.907894.00000
Central branch: y = tan(2x)-0.66291-4.00000
Central branch: y = tan(2x)-0.64219-3.39550
Central branch: y = tan(2x)-0.62148-2.94018
Central branch: y = tan(2x)-0.60076-2.58379
Central branch: y = tan(2x)-0.58005-2.29637
Central branch: y = tan(2x)-0.55933-2.05891
Central branch: y = tan(2x)-0.53861-1.85880
Central branch: y = tan(2x)-0.51790-1.68732
Central branch: y = tan(2x)-0.49718-1.53827
Central branch: y = tan(2x)-0.47647-1.40708
Central branch: y = tan(2x)-0.45575-1.29036
Central branch: y = tan(2x)-0.43503-1.18549
Central branch: y = tan(2x)-0.41432-1.09044
Central branch: y = tan(2x)-0.39360-1.00362
Central branch: y = tan(2x)-0.37289-0.92373
Central branch: y = tan(2x)-0.35217-0.84974
Central branch: y = tan(2x)-0.33145-0.78078
Central branch: y = tan(2x)-0.31074-0.71614
Central branch: y = tan(2x)-0.29002-0.65523
Central branch: y = tan(2x)-0.26931-0.59755
Central branch: y = tan(2x)-0.24859-0.54265
Central branch: y = tan(2x)-0.22787-0.49017
Central branch: y = tan(2x)-0.20716-0.43977
Central branch: y = tan(2x)-0.18644-0.39119
Central branch: y = tan(2x)-0.16573-0.34415
Central branch: y = tan(2x)-0.14501-0.29844
Central branch: y = tan(2x)-0.12430-0.25384
Central branch: y = tan(2x)-0.10358-0.21017
Central branch: y = tan(2x)-0.08286-0.16726
Central branch: y = tan(2x)-0.06215-0.12494
Central branch: y = tan(2x)-0.04143-0.08305
Central branch: y = tan(2x)-0.02072-0.04146
Central branch: y = tan(2x)0.000000.00000
Central branch: y = tan(2x)0.020720.04146
Central branch: y = tan(2x)0.041430.08305
Central branch: y = tan(2x)0.062150.12494
Central branch: y = tan(2x)0.082860.16726
Central branch: y = tan(2x)0.103580.21017
Central branch: y = tan(2x)0.124300.25384
Central branch: y = tan(2x)0.145010.29844
Central branch: y = tan(2x)0.165730.34415
Central branch: y = tan(2x)0.186440.39119
Central branch: y = tan(2x)0.207160.43977
Central branch: y = tan(2x)0.227870.49017
Central branch: y = tan(2x)0.248590.54265
Central branch: y = tan(2x)0.269310.59755
Central branch: y = tan(2x)0.290020.65523
Central branch: y = tan(2x)0.310740.71614
Central branch: y = tan(2x)0.331450.78078
Central branch: y = tan(2x)0.352170.84974
Central branch: y = tan(2x)0.372890.92373
Central branch: y = tan(2x)0.393601.00362
Central branch: y = tan(2x)0.414321.09044
Central branch: y = tan(2x)0.435031.18549
Central branch: y = tan(2x)0.455751.29036
Central branch: y = tan(2x)0.476471.40708
Central branch: y = tan(2x)0.497181.53827
Central branch: y = tan(2x)0.517901.68732
Central branch: y = tan(2x)0.538611.85880
Central branch: y = tan(2x)0.559332.05891
Central branch: y = tan(2x)0.580052.29637
Central branch: y = tan(2x)0.600762.58379
Central branch: y = tan(2x)0.621482.94018
Central branch: y = tan(2x)0.642193.39550
Central branch: y = tan(2x)0.662914.00000
Right branch: y = tan(2x)0.90789-4.00000
Right branch: y = tan(2x)0.92860-3.39550
Right branch: y = tan(2x)0.94932-2.94018
Right branch: y = tan(2x)0.97004-2.58379
Right branch: y = tan(2x)0.99075-2.29637
Right branch: y = tan(2x)1.01147-2.05891
Right branch: y = tan(2x)1.03218-1.85880
Right branch: y = tan(2x)1.05290-1.68732
Right branch: y = tan(2x)1.07361-1.53827
Right branch: y = tan(2x)1.09433-1.40708
Right branch: y = tan(2x)1.11505-1.29036
Right branch: y = tan(2x)1.13576-1.18549
Right branch: y = tan(2x)1.15648-1.09044
Right branch: y = tan(2x)1.17719-1.00362
Right branch: y = tan(2x)1.19791-0.92373
Right branch: y = tan(2x)1.21863-0.84974
Right branch: y = tan(2x)1.23934-0.78078
Right branch: y = tan(2x)1.26006-0.71614
Right branch: y = tan(2x)1.28077-0.65523
Right branch: y = tan(2x)1.30149-0.59755
Right branch: y = tan(2x)1.32221-0.54265
Right branch: y = tan(2x)1.34292-0.49017
Right branch: y = tan(2x)1.36364-0.43977
Right branch: y = tan(2x)1.38435-0.39119
Right branch: y = tan(2x)1.40507-0.34415
Right branch: y = tan(2x)1.42579-0.29844
Right branch: y = tan(2x)1.44650-0.25384
Right branch: y = tan(2x)1.46722-0.21017
Right branch: y = tan(2x)1.48793-0.16726
Right branch: y = tan(2x)1.50865-0.12494
Right branch: y = tan(2x)1.52936-0.08305
Right branch: y = tan(2x)1.55008-0.04146
Right branch: y = tan(2x)1.570800.00000
Excluded x = −π/4-0.78540-4.00000
Excluded x = −π/4-0.785404.00000
Excluded x = π/40.78540-4.00000
Excluded x = π/40.785404.00000

Draw each branch without connecting across an asymptote. A calculator’s large finite display near π/2 does not define tan(π/2): approximate entry of π can place the calculation close to the excluded angle instead of exactly at it.

A constant zero sine or cosine function has no unique least positive period. For a = 0, do not apply a non-constant wave’s fundamental-period rule. The expression 0 tan(bx) still retains tangent’s original exclusions unless a separately defined extension fills them; multiplying an undefined value by zero does not define it.

A drawing should preserve the discontinuities

Check your understanding 2 Sketch the branch with its own restrictions

Sketch the branch of y = -2 tan(3x) containing the origin, with x in radians. Mark its asymptotes, zero and the points corresponding to tangent inputs ±π/4. State its period and describe how the negative coefficient changes the branch. You may draw on paper and compare with the criteria.

Show answer

The neighbouring asymptotes are x = −π/6 and π/6, both excluded; the zero is (0,0). The points are (−π/12,2) and (π/12,−2). The branch decreases from large positive values near the left asymptote to large negative values near the right one, because the coefficient −2 reflects and stretches tangent’s output. Its period is π/3. Keep the curve separate from the vertical asymptote guides; neither guide is part of the function. These labels, point values and branch directions are the sketch-comparison criteria, without automatic marking.

Interpret parameters with units

A stated water-depth model is d(t) = 3 + 2 sin(π t/6) metres for 0 ≤ t ≤ 24 hours. The phase πt/6 is in radians. Its midline is 3 m, amplitude 2 m, range 1 to 5 m and period 12 h, since πT/6 = 2π gives T = 12 h. At t = 3 h it reaches 5 m; at t = 9 h it reaches 1 m.

The coefficient π/6 describes phase change in radians per hour, rather than a period in hours. Keep that distinction when building a model. The stated time interval bounds the model’s use; its formula does not establish future water conditions outside that interval.

Try it yourself 1

Build a model from successive extrema

Minimal support

Problem

For 0 ≤ t ≤ 12 hours, a process’s height h is modelled by a single cosine wave with successive maxima 8 m at t = 0 and t = 6 h, and a minimum 2 m at t = 3 h. Find a model h(t) = a cos(bt) + c with bt in radians and b > 0. Explain the sign and units of its constants, check the three stated extrema and find h(1.5). A proposal uses b = 6 because the period is 6 h; decide whether that can match the data.

Write your answer, then compare it with the solution

Hints

Hint 1

Separate vertical distances from a full turn of phase

Use the two extreme heights for the midline and amplitude. Then ask how much the phase must increase between successive maxima, and match the starting height before choosing the coefficient’s sign.

Show full solution

  1. Recover the vertical parameters

    Method

    Use the midpoint and half-difference of the two extreme heights.

    Reason

    The midline lies equally far from the maximum and minimum, while amplitude is that positive distance.

    Working

    Answer

    The midline is c = (8 + 2)/2 = 5 m and amplitude (8 − 2)/2 = 3 m. Since h(0) = a + c is the maximum 8 m, choose a = +3 m. A negative a would start at the minimum instead.

  2. Convert the period into a phase rate

    Method

    Use the 6 h separation between successive maxima.

    Reason

    A full cycle requires the phase bt to increase by 2π radians.

    Working

    Answer

    Thus 6b = 2π, giving b = π/3 radians per hour. The model is h(t) = 3 cos(π t/3) + 5 metres. Choosing b = 6 radians per hour would instead give period 2π/6 = π/3 h, contradicting the stated successive maxima.

  3. Check the extrema and intermediate prediction

    Method

    Substitute the stated times and t = 1.5 h into the model.

    Reason

    The height range alone cannot check the period, starting phase or behaviour between extrema.

    Working

    Answer

    At t = 0, 3 and 6 h the phases are 0, π and 2π, giving heights 8, 2 and 8 m as required. At t = 1.5 h the phase is π/2, so h = 5 m, on the descending part of the cycle. These checks agree with the stated model on 0 ≤ t ≤ 12 h; they do not prove that an unstated physical process follows it beyond that interval.

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