What it shows
A 10 cm arm pivots at the centre of a circle on a grid. At any angle, the height of its tip above the horizontal is 10 times the sine and the distance along the horizontal is 10 times the cosine, so at 30 degrees the tip is 5 cm up and 8.66 cm across.
The science
In a right triangle with hypotenuse 1, the side opposite an angle is its sine and the side next to it is its cosine. The arm is the hypotenuse of a triangle whose other sides are read straight off the grid, so sine and cosine become lengths you can measure, and tangent is the opposite divided by the adjacent. As the arm turns past 90 degrees the horizontal reading goes negative, showing why cosine is negative in the second quadrant while sine stays positive.
Materials
Cost: under Rs 50. Many items come from home or from waste; spare parts such as LEDs, motors and wire are in the store.
- A 30 cm square board
- Centimetre graph paper glued to the board
- A stiff card or aluminium strip 10 cm long as the arm
- A split pin or a bolt with a nut as the pivot
- A protractor fixed round the centre
- Thread with a bead as a plumb line
Safety
For this project: An adult must drill the pivot hole if a bolt is used.
Never use mains electricity: power models with cells, batteries or a low-voltage adapter. Sharp and hot tools, flames and lab chemicals need an adult or your teacher beside you. Keep magnets, button cells and small parts away from young children, and wash your hands after making.
All the safety rulesStep by step
8 steps. Do them in order, and test each part before you go on to the next. Write what you notice in a notebook as you go.
Glue graph paper on the board and draw axes through the centre, with 1 cm standing for 0.1 unit so the circle of radius 10 cm is the unit circle.
Draw the circle of radius 10 cm and fix the protractor so 0 degrees lies along the positive x axis.
Pivot the arm at the centre with the split pin; mark its tip clearly and make sure its length from pivot to tip is exactly 10 cm.
Set the arm to 30 degrees and read the tip: height 5.0 cm and horizontal 8.7 cm, so sin 30 = 0.50 and cos 30 = 0.87.
Repeat for 45 and 60 degrees and record the readings beside the table values 0.71, 0.71 and 0.87, 0.50.
Turn the arm to 150 degrees and show that the height is still 5 cm but the horizontal reading is on the negative side.
Hang the plumb thread from the tip so visitors see the vertical drop that is the sine.
Check that for every angle the square of the height plus the square of the horizontal reading is 100, which is sin squared plus cos squared equals 1.
What you should see
The board reads sin 30 as 0.5, cos 30 as about 0.87, and sin 45 and cos 45 both about 0.71, and shows cosine turning negative past 90 degrees.
Viva questions
A teacher or a judge will ask how it works. Try to answer each question before you open it, then compare.
1Why is the circle radius 10 cm called a unit circle?
Because 10 cm on the board stands for 1 unit, so the arm length is 1 and its projections are the sine and cosine directly.
2What is sin 90 degrees on the board?
1, the tip is straight up 10 cm, and cos 90 is 0.
3Why is sin squared plus cos squared equal to 1?
By Pythagoras on the triangle with hypotenuse 1.
4In which quadrant are both sine and cosine negative?
The third quadrant, angles between 180 and 270 degrees.
Take it further
Add a vertical tangent line at x = 10 cm and extend the arm to meet it: the height where it meets is 10 times tan of the angle.
Change one thing, measure what happens, and add a table or chart of your readings: that is what turns a model into your own investigation.
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