Question

In physics it is important to use mathematical approximations. Demonstrate that for small angles (< 20°) the following relationship is true where α is in radians and α' is in degrees.

tan(α)　~ sin(α) ~ α = (pi*α')/(180 deg)

Use a calculator to find the largest angle for which tan α may be approximated by α with an error less than 10.0%.

Answer 1

note that
πxa / 180° = a

Answer 2

I don't understand. :(

Answer 3

well pi, when measured in radians is equal to 180° hence

\[(\pi \times \alpha)/ 180 = \alpha,\]
this is just a straight line graph cutting the origin at the angle alpha.

infact all three lines cut the origin (try inserting \alpha\ = 0 all equations give 0

Answer 4

The Yellow line is tan(alpha)
the red line is sin(alpha)
the black line is alpha

When we are close to the origin these lines are on top of each other, ie a good approximation of eachother

Answer 5

thank you :)

Answer 6

see how sin and tan both cut the x-axis when x is ± π (in radians) or at 180°

the lines are all in close approximation between ±0.6 x ~ 20°