How to convert 1000° into radian unit in order to find the corresponding point on the trigonometric circle. Is there a general formula to do this?

Question

My "understanding" of the radian concept dates back to a few days.

I'm trying to convert a 1000° angle to radians, and to find the corresponding point on the trigonometric circle.

I'm thinking of this.

(1) How many times do I go aroud the circle from 0 to 2 Pi ( in the positive sense) ?

1000/360 = 2, 77777777778

So : 2 times

I'm left with an angle of O,77777777778 times 360 = 280°

(2) How many radians in 280 °

• One radian is : 360/(2pi).

• So : 280° in radians is 280/ [ 360/(2pi) ] = 4,8869...

(3) How many "PiRadians" in 4,8869... radians ?

4,88,69 radians in " PiRadians" is : 1,55555555556 PiRadians

(4) 1, 55 Pi Radians is not far from 15/10 Pi Radians = 3/2 Pi Radians

So a 1000° angle is approximately an angle of : 3/2 Pi Radians

(5) If I am correct, cos(1000) is not far from 0, since

                 cos ( 3/2 Pi) = 0. 

(6) BUT the calculator tells me that cos(1000) = O,1736...

What did I miss?

Answer 1

To convert degrees to radians, you can simply multiply by $\frac{\pi}{180}$, and then subtract multiples of $2 \pi$. In this case, you’d get $\frac{50 \pi}{9}$ which would be the same as $\frac{14 \pi}{9}$. It is true that this is not too far from $\frac{3 \pi}{2}$, but there is still some non-zero error which yields the offset from the calculator.

Answer 2

$1000^{o} = 900^0 + 100^o = 5\pi+\frac{100\pi}{180}$(radians)

Now, $\cos((2n-1)\pi+\theta) = -\cos\theta$ [here $2n-1 = 5$]

So, $\cos(1000^{o}) = \cos(5\pi+\frac{100\pi}{180}) = -\cos(5\pi+\frac{5\pi}{9}) = -\cos(\frac{5\pi}{9}) = +0.1736\cdots$