# Trigonometric Graphs and Applications | Maths with Lemon

Source: https://mathswithlemon.io/aa-geometry-trigonometric-graphs-applications.html


### Graphs and Applications of Trigonometric Functions

IB Mathematics: Analysis and Approaches · Lessons 5.4 and 5.5 · AHL 3.7. Explore periodic graphs, transformations, equations from graphs and mathematical models of real-world motion.


#### The Three Parent Graphs


##### What you need to know

- Exact values from the unit circle
- Angles in degrees and radians
- Coordinates, intercepts and asymptotes
Review Using the Unit Circle before graphing.


##### Key Points

The sine and cosine graphs repeat every \(2\pi\) radians. Tangent repeats every \(\pi\) radians and has vertical asymptotes.

Range: \([-1,1]\). Starts at the origin.

Range: \([-1,1]\). Starts at a maximum.

Period: \(\pi\). Undefined at \(x=\frac{\pi}{2}+k\pi\).

Amplitude: distance from midline to peak. Period: horizontal length of one cycle. Midline: central horizontal line.


#### Transformations of Sine and Cosine


##### What you need to know

- Vertical stretch and reflection
- Horizontal stretch and compression
- Phase shift and vertical translation
- Sketch one full cycle using five key points

##### Key Points

Use the transformed function to read off its graph features.

For sine and cosine, maximum and minimum values are \(k+|a|\) and \(k-|a|\).

Consider \(y=2\sin\left(3\left(x-\frac{\pi}{6}\right)\right)+1\).

Maximum = 3, minimum = −1. The five key points of one cycle are spaced \(\pi/6\) apart.


#### Interactive Trigonometric Graph Explorer


##### Try this

- Choose sine, cosine or tangent
- Adjust amplitude / vertical scale, frequency, phase shift and midline
- Observe how the period changes
- Compare your sketch with the curve
For tangent, the vertical scale is not an amplitude: tangent has no maximum or minimum.


##### Explore transformations


#### Graphing Tangent


##### What you need to know

- Locate vertical asymptotes
- Find the tangent period
- Identify its central crossing
- Distinguish vertical scaling from amplitude

##### Key Points

The central crossing occurs at \((h,k)\). Vertical asymptotes occur where the tangent argument equals \(\frac{\pi}{2}+n\pi\).

For For the function \(f(x)=3\tan\left(x-\frac{\pi}{4}\right)+1\), the period is \(\pi\), and the central crossing is \(\left(\frac{\pi}{4},1\right)\). , and the closest asymptotes are:

For the function \(f(x)=3\tan\left(x-\frac{\pi}{4}\right)+1\), the period is \(\pi\), and the central crossing is \(\left(\frac{\pi}{4},1\right)\).


#### Writing an Equation from a Graph


##### Strategy

- Find the maximum and minimum values
- Calculate the midline and amplitude
- Measure the period
- Choose sine or cosine based on a convenient starting point
- Check the phase shift
Practice with the ten graph sketches on page 2 of Lesson 5.5.


##### Key Points

A curve has maximum 5, minimum −1 and period \(\pi\). It starts at its maximum when \(x=0\).

A suitable model is:

Different but equivalent sine or cosine equations may describe the same graph.


#### Applications: Ferris Wheels and Paddle Wheels


##### From the worksheets

- Ferris-wheel height and rotation
- Riverboat paddle-wheel motion
- Maximum and minimum heights
- Times when a specified height is reached
Lesson 5.4, pages 5–6, and Lesson 5.5, pages 3–5.


##### Model 1 — Paddle wheel (Lesson 5.5)

The worksheet gives the height in feet, with time in seconds:

8 ft

4 seconds

6 ft

−2 to 14 ft

At the waterline, \(h(t)=0\). Therefore:

Solve with a calculator, then select times in the requested interval.


##### Model 2 — Navy Pier Ferris wheel

Its amplitude is 70 ft, period is 7 minutes, and heights range from 10 ft to 150 ft. Lesson 5.5 asks when the car reaches 100 ft and 0 ft; the latter is impossible in this model.


#### Applications: Tides and Periodic Data


##### Interpret the model

- Amplitude describes variation from average water depth
- Midline gives average depth
- Period describes the time between matching high tides
- Compare models before drawing conclusions
See Lesson 5.5, page 6.


##### Model from Lesson 5.5 — Maine coastline

Here \(d\) is the water depth in feet, and \(t\) is the time in hours after the first high tide.

The model predicts a maximum depth of 13.15 ft and a minimum of 4.43 ft.


#### Extra: Worksheets and Revision


##### Material and references

- Lesson 5.4 — Graphs of Trigonometric Functions
- Lesson 5.5 — Applications of Trigonometric Functions
- Unit Circle
- AA Geometry & Trigonometry
- IB Calendar
Upload both PDFs alongside this HTML file, or update the links to their actual folders on your server.


##### Revision checklist

Sketch sine, cosine and tangent over one period.

Read amplitude, period, phase shift and midline.

Find a model using key points.

Interpret period and amplitude with units.

Recommended order: Lesson 5.4 pages 1–4, then Lesson 5.5 page 2 and modelling problems on pages 3–6.


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