Use the exact values of the sin, cos and tan of pi/3 and pi/6, and the symmetry of the graphs of sin, cos and tan, to find the exact values of si -pi/6, cos 5/3pi and tan 4pi/3.

I have found the answers to the first three using the special tables
sin(ƒÎ/6) = cos(ƒÎ/3) = 1/2
cos(ƒÎ/6) = sin(ƒÎ/3) = �ã3/2
tan(ƒÎ/3) = �ã3
tan(ƒÎ/6) = 1/�ã3
I know that the symmetry of sin(-pi/6 is in the third quadrant, so 7pi/6 =
- 0.5
the symmetry of cos 5pi/3 i have taken 2pi - 5pi/3 to give cos pi/3 = 0.5 but i am not too sure where the symmetry comes in
With the tan I do not quite know where to start. Maybe pi + pi/3.
As for plotting the graphs of sin,cos and tan; I do not know where to start here using the pi radians (whereas as usually I am good at plotting graphs. (perhaps it is an age thing :-(

Your symbols baffle me. Sorry.

what symbols

You have been using alt-251 for square-root, which probably works for Microsoft Word, but not for this forum which requires HTML codes.

For π, type in
& p i ;
(without the spaces in-between) and for
square-root, use
& r a d i c ;

To find the exact values of sin(-pi/6), cos(5pi/3), and tan(4pi/3), we can use the symmetry properties of the trigonometric functions.

1) sin(-pi/6):
Since sin(x) is an odd function, it exhibits symmetry about the origin. This means that sin(-x) = -sin(x) for any angle x. Therefore,
sin(-pi/6) = -sin(pi/6) = -1/2

2) cos(5pi/3):
To determine the value of cos(5pi/3), we can utilize the symmetry property of the cosine function. Cosine is an even function, which means that cos(-x) = cos(x) for any angle x. Therefore,
cos(5pi/3) = cos(-pi/3) = cos(pi/3) = 1/2

3) tan(4pi/3):
To find the value of tan(4pi/3), we need to consider the symmetry of the tangent function. Tangent is an odd function, so tan(-x) = -tan(x) for any angle x. Therefore,
tan(4pi/3) = -tan(-(2pi - pi/3)) = -tan(2pi - pi/3) = -tan(pi + pi/3)
At this point, we have converted 4pi/3 to an equivalent angle in the interval [0, 2pi] using the fact that 2pi = 2pi + 4pi = 6pi. Now, since the tangent has a periodicity of pi, we can further simplify it as:
tan(pi + pi/3) = tan(pi/3) = sqrt(3)

Regarding plotting the graphs of sin, cos, and tan using radians, here are a few tips:

- For the sine and cosine functions, one full cycle is completed in the interval [0, 2pi]. Therefore, you can plot the values of sin(x) and cos(x) for x ranging from 0 to 2pi, using regular intervals like pi/6 or pi/4.

- For the tangent function, it has vertical asymptotes every pi units and repeats its pattern every pi units as well. So, a good range for plotting tan(x) would be [-pi/2, pi/2] and then replicate the pattern every pi units outside of this interval.

With these guidelines, you can plot the graphs by calculating the values of sin(x), cos(x), and tan(x) at different intervals of x and then connecting the points to create the curves for each function.