MATH TRIGONOMETRY qUESTION?????

Question

find the exact value

sin(2 pi/3)

I forgot how to do these. can someone please explain

Answer 1

sin (2π / 3) = √3 / 2 (in 2nd quadrant so is +ve)
Also , 2π/3 = 120° thus:-
sin 120° = sin 60° = √3 / 2

Sin | All
--------
Tan|Cos--------may ring a bell?

Answer 2

The conversion factor is that 180 degrees = pi radians
So 2 pi / 3 = 120 degrees

Now considering that the sine wave goes from 0 at 0 degrees to 1 at 90 degrees, to 0 at 180 degrees, and is symmetrical about the 90 degree value, then sin 120 degrees, which is sin (90+30) degrees, will also be the same as sin (90-30) degrees, i.e. sin 60 degrees.

A 60 degrees angle can be obtained in an equilateral triangle, with three sides of 2 units each. If a perpendicular is dropped from the apex to the base, you will have 2 right-angled triangles, with the base angles being 60 deg and 90 deg, and the apex angle being 30 deg. Also, in this process the base of the equilateral triangle is divided exactly into half, so the length of the base in the right-angled triangle is 1 unit.

Thus, with a hypotenuse of 2, and a side of 1, using Pythagoras theorem, the length of the perpendicular is squareroot (2^2 - 1^2) = squareroot(3).

Since sine = opposite/hypotenuse, then sine 60 will be squareroot(3)/2

Answer 3

If you want to think in terms of degrees which is easier for some people, use pi=180

So 2 pi/3 = 120 This is in the second quadrant so the sin is positive and equal to SQRT(3)/2

Know your quandrants and what sign the trig functions have. And to think of these in terms of basic triangles (see further down).

First quadrant (0 to 90 degrees): sin, cos and tan are all positive.
Second quadrant (90 to 180 degrees). Sin positive but cos and tan are negative.
Third quadrant (180 to 270 degtrees). sin, cos and tan are all negative.
Fourth quadrant (270 tp 360 degrees). Cos is positive and sin and tan are negative.

Also you need to know trig values for some of the more commonly used angles. This can be done either memorize the values of memorize the triangles from which you can derive the values.

30 (and 60 degrees, different angles in the same right triangle). this triangle has sides SQRT(3) and 1 with a hypotenuese (probably spelled that wrong) of 2. The 30 degrees is between the 2 and SQRT(3) so the sine of 30 degrees is the SQRT(3)/2, and so on.

45 degrees has sides 1, 1 and SQRT(2)

Answer 4

you should really study the unit circle. Sin (2pi/3) is the same thing as the y coordinate at 2pi/3 which is square root of 3 divided by 2. (cant find the square root symbol) I know it is a bit tedious, but really get to know the unit circle in degree and radian measure and also the x and y coordinates at each point. It really really really makes life easier later on.

Answer 5

I am going to assume you can use a calculator. Pi is 3.141592654 on your calculator. Multiply that by two and divide that by three. Set your calculator to radians. Then enter that number.

Another way to think of it is that there are two pi radians in a complete circle. One third of that is 120 degrees. Put that in to your calculator.

Answer 6

sin (2pi/3) = (sqrt3)/2

pi/3 has coordinates (1/2, sqrt3/2)
and it changes its sign whenever you change your quadrant.. since the 2pi/3 is on the 2nd quadrant, your cos will be - and your sin will still be +.

Answer 7

sin (2pi/3)=sin(2(180)/3)=sin(120) =0.866