How to find sin(400pi/3) using identities?

Answer 1

sin(400π/3)=sin(399π/3+π/3)=sin(133π+π/3)

=sin(133π)cos(π/3)+sin(π/3)cos(133π)

=0*(1/2)+((√3)/2)*(-1)

Answer 2

The angle 400π/3 radians is unfamiliar because it is so big!

The much smaller angle 0 radians (i.e., 0 degrees) is easy to work with because we know its sine and cosine:

sin 0 = 0

cos 0 = 1.

Other ways to write this same angle of 0 radians would be 2π radians, 4π radians, 6π radians, or even 132π radians, because whenever we add 2π radians (i.e., 360 degrees) to an angle, we end up with the same angle we started with. Note that the numbers 2, 4, 6, and 132 are all even.

I hope I have convinced you that

sin (132π) = sin 0 = 0

cos (132π) = cos 0 = 1

Hey, 132π radians and 0 radians are the very same angle!

Let's do some simple arithmetic without the π: 400/3 = 133 + 1/3.

But 133 is an odd number, and I'd rather have an even number, so let's write this as 400/3 = 132 + 4/3.

Now let's put the π back in.

Therefore our angle 400π/3 radians = 132π radians + (4/3)π radians.

They want us to use a "trig identity" such as the flagship identity

sin(x + y) = (sin x)(cos y) + (cos x)(sin y).

So, using this identity and letting x=132π radians and letting y=(4/3)π radians, we have

sin (400π/3) = sin (132π + (4/3)π) = (sin 132π)(cos (4/3)π) + (cos 132π)(sin (4/3)π)

= 0·(cos (4/3)π) + 1·(sin (4/3)π)

= sin ((4/3)π)

Now what is sin ((4/3)π radians)?

It's easier to talk about this angle in degrees.

(4/3)π radians = 240 degrees, which is in the third quadrant.

[Footnote. How did I know that (4/3)π radians = 240 degrees? It's because 2π radians is a complete circle, so (4/3)π radians is 2/3 of a complete circle (because 4/3 is 2/3 of 2), and 2/3 of a complete circle in degrees is 240 degrees.]

In this quadrant, sine is negative.

Draw a picture of this angle pointing to the lower left at 7:00 o'clock in a circular clock face.

Instead of making the radius of the clock face be 1 unit long, let the radius be 2 units.

Now draw a 30-60-90 right triangle by drawing one straight line from the 7:00 point on the circumference straight up to the X axis. The line you just drew has a length of √3, and the hypotenuse has a length of 2.

sin (240 degrees) = opposite/hypotenuse = -(√3)/2

So sin (400π/3 radians) = sin ((4/3)π radians) = sin (240 degrees) = -(√3)/2

Answer 3

sin(400pi/3) = sin(400pi/3 - any multiple of 2pi)

= sin(400pi/3 - 396pi/3)

= sin(4pi/3)

= -sin(pi/3)

= -sin60

=- -sq3/2

= about - 1.732/2

= about - 0.866

Answer 4

sin(400π/3) = sin(132π + 4π/3) = sin(4π/3) = - sin(π/3) = - √3/2