how can i get the trigonometric functions in fractions without using a calculator? for ex: -sin 4pie/3?

Question

Im having trouble doing trigonometric functions without a calculator. I want to get the answer in fractions not in decimals. Is there an easy way? if so can u please show some examples as well? greatly appreciated!

Answer 1

Angle___cos___sin___tan
π/3_____1/2___√3/2__√3
4π/3__ - 1/2__-√3/2__√3

sin 4π/3 = - √3/2

Signs are allocated according to:-
sin | All
---------
tan| cos

4π/3 lies in 3rd quadrant so sin 4π/3 is - ve

Answer 2

First, only consider the first quadrant: 0 to π/2.

There are 5 special angles - 0, π/6, π/4, π/3, π/2, which respectively equals 0˚ ,30˚, 45˚, 60˚, 90˚

For sine, their values are
0, (√1)/2,(√2)/2,(√3)/2 and 1=(√4)/2
For cosine, their values are
1=(√4)/2,√3/2,√2/2,√1/2, and 0

When determining the values for other quadrants, find the reference angle from the x-axis, i.e. 135˚ uses 45˚ and 330˚ uses 30˚. Apply the trig functions on the reference angles to fine the values.

Now use the CAST method to recall the sign for each quadrant:

S | A
---------
T | C

Where A means all are positive, S means only sine is positive, C for cosine, and T for tangent. Throw in a negative sign for the rest.


* Disclaimer: other methods may be introduced in your book, so it is always best to refer back to your own textbook

Answer 3

I remember using some kind of table for that when I was in school...