What are exact ratios in trigonometry?

Hi. I have a task: Using exact ratios, show that
sin^2 60*sec60* - cosec60*= (9-4√3)/6

Could please explain what "Using exact ratios" means?

Cheers

Hi. I have a task: Using exact ratios, show that
sin^2 60*sec60* - cosec60*= (9-4√3)/6

Could please explain what "Using exact ratios" means?

Cheers

Don't even get me started on exact trig values. They really interest me for some reason. You can find all sine values for integer angles (degrees). Even $\sin 1^{\circ}$ can be found exactly and from there you can find them all.
But yeah for these questions just use the exact value of sin(60) and cos(60) and you'll have to rationalise the denominator along the way until you get it into the very attractive form on the right hand side as you posted.

Have you read through ptolemy's construction of trigonometric tables. I believe that's where we get Ptolemy's Theorem from.

Even $\sin 1^{\circ}$ can be found exactly.

Here it is.



Unfortunately, the answer involves the cube roots of complex numbers. If you are looking for 'nice' algebraic forms for trigonometric functions in degrees you need to stick to the multiples of 3.