Lesson 28 Graph of the rotating arm

# Graph of the rotating arm

Overview Resources
This is an activity on the sine and cosine ratios as functions of angle size. The dynamic meaning of angle as a turn is demonstrated directly, unlike in tables of values, or frozen values in a right-angled triangle. The angle is demonstrably used as a variable, on which depends another variable, this time a ratio of sides. The visual context allows explorations of the formal mathematical models.

The main issues are the differences between the trigonometric ratios in triangles, and these ratios as functions of angle of turn.
Worksheets 1 and 2. Calculators with sine function. Tracing paper

Rotating arm - large card or hardboard circle, radius 10 units (approx. 30-40 cm), with angles marked in 10° intervals. Horizontal diameter drawn in. Two ‘arms’ made of card or plastic the same length as the radius, marked in 10 equal intervals. These arms are joined by a split pin. One arm is attached at the centre of the circle with another split pin.
Exploring the concepts underlying trigonometry. Use of sine, cosine and tangent in right-angled triangles.

Follows on from TM25:Triangle ratios.
EPISODE 1
Sine ratios within a circle plotted on a graph (up to 180°)
The rotating arm apparatus is used to demonstrate the dynamic and static aspects of angles. Using a circle with a radius of 10 units, pupils find the sine ratio by drawing the position of a rotating arm at different angles and finding the vertical projection from the points on the circumference. They plot these values on a graph and explore the symmetry of the graph. They discuss how the vertical projection reaches its maximum value at 90° then declines back to 0.
EPISODE 2
Plotting the sines beyond 180°
Pupils continue with angles up to 360°. They discuss how to represent the vertical projection when it is below the horizontal reference level. They continue the plotting of the function and discuss the pattern of the graph, finding their own descriptions of the sinusoid, They consider the meaning of angles greater than 360° and less than 0°.