exact trigonometric values

this diagram conveys a lot of information

you can find sin, cos and tan of 30, 60 and 45 degrees in surd (exact) form
but you can also, with a bit of work, find sin, cos and tan of 15 and 75 degrees in surd form

simplification requires rationalising the denominator















given some information about tan 15 and tan 75, you can find them in surd form:

triangular prisms

a powerpoint for this is here

perpendicular to the hypotenuse

thanks to 'five triangles' on twitter  (on jan 7th 2016)

18 circles

which circle packing arrangement is the most compact?
an opportunity to use exact forms of trigonometric ratios

Paul Lockhart's problems

these are from Paul Lockhart's 'measurement' book

finding relationships between the radii, between a radius and the side length of a square (first sheet) or the length of an equilateral triangle (third sheet) can and probably ought to involve:

• symmetry
• exact trigonometric values
• surd manipulation

the first sheet doesn't need trigonometry, just pythagoras in an isosceles right angled triangle

you do not need to manipulate surds, but you obtain simpler looking solutions if you do

the second sheet involves expanding brackets and some not too hefty algebra, to obtain a neat result

the third sheet involves exact trigonometric values for 30 and 60 degree angles

solutions
well, I think they are anyway...

exact trigonometric values

a ppt is here

find a quick(er) way to do this question

sine rule

a powerpoint for this