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 sin
-pi/6, cos 5/3pi and tan 4pi/3.
I have found the answers to the first three using the special tables
sin pi/6 = cos pi/3 = 1/2
cos pi/6 = sin pi/3 = �root 3/2
tan pi/3 = �root 3
tan pi/6 = 1/�root 3
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 :-(
tan 4 pi/3 is in quadrant three pi/3 or 60 degrees below the -x axis
so
both x and y are negative
tan (pi+pi/3) = tan (60 deg) = sqrt(3)
Can anyone give me some pointers on how to plot the graph in excel please
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 the sine function is an odd function, we know that sin(-x) = -sin(x). Therefore, sin(-pi/6) = -sin(pi/6) = -1/2.
2. cos(5pi/3):
To find the value of cos(5pi/3), we can use the fact that the cosine function is an even function, which means cos(-x) = cos(x). Since 5pi/3 is in the third quadrant, we can use the symmetry property to find the equivalent angle in the first quadrant. 2pi - 5pi/3 = pi/3. Therefore, cos(5pi/3) = cos(pi/3) = 1/2.
3. tan(4pi/3):
To find the value of tan(4pi/3), we first note that the tangent function is an odd function, tan(-x) = -tan(x). Therefore, tan(4pi/3) = -tan(-4pi/3). To find the equivalent angle in the first quadrant, we can add 2pi to -4pi/3 to get -4pi/3 + 2pi = 2pi/3. So, tan(4pi/3) = -tan(2pi/3) = -√3.
As for plotting the graphs of sin, cos, and tan using pi radians, here are some guidelines:
1. For the sine function, we know that sin(0) = 0, sin(pi/6) = 1/2, sin(pi/3) = √3/2, sin(pi/2) = 1, sin(2pi/3) = √3/2, sin(5pi/6) = 1/2, and sin(pi) = 0. These points can help in plotting the graph of the sine function.
2. For the cosine function, we know that cos(0) = 1, cos(pi/6) = √3/2, cos(pi/3) = 1/2, cos(pi/2) = 0, cos(2pi/3) = -1/2, cos(5pi/6) = -√3/2, and cos(pi) = -1. These points can help in plotting the graph of the cosine function.
3. For the tangent function, we know that tan(0) = 0, tan(pi/6) = 1/√3, tan(pi/3) = √3, tan(pi/2) is undefined (as it corresponds to an asymptote), tan(2pi/3) = -√3, tan(5pi/6) = -1/√3, and tan(pi) = 0. Plotting these values can help in drawing the graph of the tangent function.