Sinusoidal Models (AI SL)
Tides, temperatures, wheels and anything else that rises and falls in a repeating pattern can be modelled with a sine or cosine function. This page covers how to read off amplitude, period and midline directly from the model, plus how to solve for a given output value on your GDC. It's part of the broader Other Models topic.
12 questions on this sub-topic.
The two forms
Covered under IB syllabus reference SL2.5: modelling with sinusoidal functions \(f(x)=a\sin(bx)+d\) and \(f(x)=a\cos(bx)+d\), with the angle in degrees. You find amplitude \(a\), period \(\tfrac{360}{b}\), or the equation of the principal axis \(y=d\), but you are not expected to convert between the sine and cosine forms.
Sinusoidal model (sine form)
\(f(x)=a\sin(bx)+d\)
Amplitude \(|a|\), period \(\tfrac{360}{b}\), principal axis \(y=d\).
Sinusoidal model (cosine form)
\(f(x)=a\cos(bx)+d\)
Same reading: amplitude \(|a|\), period \(\tfrac{360}{b}\), midline \(y=d\).
Need the full syllabus wording and modelling context? See Other Models.
Worked examples
The temperature (°C) in a city is modelled by \(T(t) = 15 + 8\sin(30t^\circ)\), where \(t\) is the time in hours after midnight.
(a) State the maximum temperature.
(b) Find the temperature when \(t = 3\).
Worked solution
(a) Maximum temperature. The sine term ranges over \([-1,1]\); the maximum of \(T=15+8\sin(30t^\circ)\) is when \(\sin=1\): \(T_{\max}=15+8(1)=23\text{°C}.\) A1
(b) Temperature at \(t=3\). The angle is \(30\times3=90^\circ\), and \(\sin90^\circ=1\): \(T(3)=15+8\sin(90^\circ)=15+8=23\text{°C}.\) A1
The depth model is \(D(t)=5+3\sin(30t^\circ),\ 0\le t\le12.\)
(a)(i) Find the earlier time when the depth is 7 m, to 3 significant figures.
(a)(ii) Find the later time, to 3 significant figures.
Worked solution
(a)(i) \(t\approx1.39\) h A1
(a)(ii) and \(t\approx4.61\) h A1 (these are symmetric about the peak at \(t=3\)).
The temperature is \(T(t)=18+6\cos(15t^{\circ})\,^{\circ}\text{C}\), \(t\) hours after noon.
(a)(i) State the maximum temperature.
(a)(ii) State the minimum temperature.
(b) Find the temperature at \(t=6.\)
Worked solution
(a)(i) Max. Amplitude \(6\) about midline \(18\): max \(=18+6=24\)°C. A1
(a)(ii) Min \(=18-6=12\)°C. A1
(b) Temperature at \(t=6\). Angle \(=15\times6=90^\circ\), \(\cos90^\circ=0\):
\(T(6)=18+6\cos(90^\circ)=18+0\) M1
\(=18\text{°C}.\) A1
Common mistakes
- Leaving the GDC in radian mode. AI sinusoidal models are written with the angle in degrees, like \(\sin(30t^\circ)\) - a calculator left in radian mode gives a completely different, wrong value with no warning.
- Reading off the wrong quantity. The amplitude is \(a\), not the maximum value - the maximum is \(d+|a|\) and the minimum is \(d-|a|\). Mixing these up is a common slip when a question asks for "the maximum" rather than "the amplitude".
- Only finding one solution when two exist. A sine curve usually crosses a given height twice per cycle; if a question expects two answers over an interval, check the graph for a second intersection rather than stopping after the first.
Ready to practise properly?
19 sinusoidal-model questions, marked instantly like the real exam.
Quick answers
How do you find the amplitude of a sinusoidal model?
In \(f(x)=a\sin(bx)+d\) or \(f(x)=a\cos(bx)+d\), the amplitude is \(|a|\), the coefficient multiplying the sine or cosine term.
How do you find the period of a sinusoidal model?
The period is \(\tfrac{360}{b}\) when the angle is measured in degrees, which is how AI sinusoidal models are usually written.