how to find exact answer for a radiant circle?

Question

without looking at the unit circle in math, how can you mathematically find out the answer?

i.e : sine(theta)=1/2

how do you mathematically figure out the answer that it's Pi/6 ?

or cos(pi/3)=1/2


and so on

please help

Answer 1

for angles 0,30,45,60,90 deg (0,pie/6,pie/4,pie/3 & pie/2 rad. respect.) the values should be known just like before doin any trigno. ques. these form. be on ur tips,
sinx=oppo. side/hypoten. e.t.c.now coming back to ur ques. sin(theta)=1/2
theta=sin-1(1/2) [here sin-1=sin inverse)
theta=sin-1(sin pi/6)
theta(in rad.)=pie/6

Answer 2

theta varies between 0 and 90 degrees

it would make sense that sin 30 = 1/2

Pi in radians is equal to 180 degrees.
therefore 180/3 = 60
cos 60 = 1/2

cos is the opposite of sin, therefore cos 0 = 1 whereas cos 90 = 0

You should just draw a sin and cos graph to see what is happening for yourself.

Answer 3

in radians, pi = 180 degrees. this all stems from the idea that the circumference of a circle represents 360 degrees and we know that circumference = 2 * pi * radius

Answer 4

those things should be memorized, but if u don't want to, or not CAPABLE of,,,,, u should memorize it, 'cause these angles are called POPULAR ANGLES which are

0, pi/6,pi/4,pi/3,pi/2,pi,,,, and their multiplies with integers