How do I write 30 ˚ and 45 ˚ with fraction?
You understand, don't you, that your initial post said nothing about sine and cosine?
For 45 degrees think of an isosceles right triangle. If the two sides are both of length 1, then, by The Pythagoream theorem, the hypotenuse has length $\displaystyle \sqrt{2}$. So $\displaystyle sin(45)= cos(45)= \frac{1}{\sqrt{2}}= \frac{\sqrt{2}}{2}$.
For 30 or 60 degrees, start with an equilateral triangle, with sides of length 1. Drop a perpendicular from one vertex to the opposite side. Since an equilateral triangle has angles of measure 60 degrees, this divides the triangle into two right triangles with angles of 30 and 60 degrees. The hypotenuse of the right triangles is 1 and the leg opposite the 30 degree angle is 1/2 that side of the equilateral triangle and so has length 1/2. By the Pythagorean theorem again, the other leg has length $\displaystyle \sqrt{1- (1/2)^2}= \frac{\sqrt{3}}{2}$.
Thus, $\displaystyle sin(30)= \frac{\frac{1}{2}}{1}= \frac{1}{2}$. $\displaystyle cos(30)= \frac{\frac{\sqrt{3}}{2}}{1}= \sqrt{3}{2}$, $\displaystyle sin(60)= \frac{\frac{\sqrt{3}}{2}}{1}= \sqrt{3}{2}$, and $\displaystyle cos(60)== \frac{\frac{1}{2}}{1}= \frac{1}{2}$.
In a right triangle:
$\displaystyle
\sin A = \dfrac{ \text{opposite}} {\text{hypotenuse}}
$
$\displaystyle
\sin (30deg) = 0.5 = \dfrac{ 1 } { 2 }
$
$\displaystyle
\sin (45deg) = 0.707106781 = \dfrac{ 1 } { \sqrt{ 2 } }
$
also
$\displaystyle
\cos A = \dfrac{ \text{adjacent}} {\text{hypotenuse}}
$
Is that what you are seeking?
.
Well, i had so many probles, with english, that i forget.Originally Posted by HallsofIvy
What I am asking is how can I calulate the value of any cos and sin.
When there is cos180, for example, i look here(http://www.cartage.org.lb/en/themes/...unitcircle.gif) and i can see that this is minus one. But i dont know how to calculate the values of 15,70,120,150,.... . When the value is diffrent than 90,180,270, i dont know how to calulate the value of sinus and cosinus.
And soory again for my bad english.
But for general angles, "When the value is diffrent than 90,180,270, i dont know how to calulate the value of sinus and cosinus", the only thing to do is use a calculator! The ones I gave, 30, 45, and 60, your 90, 180, 27, and angles that, as Mr Fantastic says, can be reduced to those, are the only angles for which exact values of sine and cosine can be calculated.