Well, cos(7pi/6) is the cosine of the angle represented by (7pi)/6.
This angle would be (6pi)/6 + (1pi)/6.
(6pi)/6 is obviously just pi, and, from the unit circle, we know that the angle pi radians is 180 degrees.
Think of a circle, with the center lying on the origin. We would start at the right hand side of the circle, on the x-axis, and when we move we would move counterclockwise. So after going 180 degrees, we would be on the left hand side of the circle, again on the x-axis.
Now we add to that (1pi)/6. This angle, in degrees, would be 30, and we can figure this out, if we need to, by remembering that 2pi is 360 degrees. So we get pi/6 if we divide 2pi by 12, so if we divide 360 by 12, we get 30, so pi/6 radians is 30 degrees.
So, continuing in the same counterclockwise path, we would move around the circle the distance of 30 more degrees.
We are now on a specific point on the circle. We can imagine that we draw a vertical line to the x-axis from this point, a line from there on the x-axis to the origin, and another straight line from our point on the circle to the origin. Now we have a triangle.
The angle of this triangle at the corner made by the line on the x-axis and from the origin to the point on the circle would be 30 degrees, right? Because that’s 30 degrees down from the x-axis to the point, is what we had already decided. The angle made by the vertical line from the point on the circle to the x-axis and the line from there to the origin would be a 90 degree angle, right? Because a vertical line to the x-axis, and a line following along the x-axis, would be perpendicular. That’s 90 + 30 = 120 degrees, and every triangle has 180 degrees, so the last angle is 180 - 120 = 60 degrees.
What’s this? A 30-60-90 triangle! A triangle with these angles has side lengths that are known to be (as a ratio):
Opposite the 30 degree angle: 1
Opposite the 60 degree angle: square root of 3
Opposite the 90 degree angle: 2
You are asked to find cos(7pi/6). We are doing this by finding cos(pi/6), or cos(30 degrees). The cosine is the ratio of the adjacent side to the hypotenuse. The hypotenuse, the longest side, would be, of the 1, square root of 3, and 2, the 2. The side opposite the 30 degree angle is 1, so the side that’s left, the adjacent side to the 30 degree angle, is the square root of 3. So the cosine of 30 degrees is, adjacent over hypotenuse, or the square root of 3 over 2.
But we’re not actually done. Because you were asked to find cos(7pi/6), not cos(pi/6). Remember that our triangle is in the third quadrant, so all the points of the ends of the lines that make up this triangle have negative values. So it’s not really the square root of 3 over 2, but the negative square root of 3 over 2.
That’s long and probably convoluted, but a summary of the points of the process for these kinds of questions is:
Imagine a circle, and find a point on that circle that’s however many degrees away from the right hand side, going counterclockwise.
Draw a vertical line from that point to the x-axis, and a line from there to the origin, and a line from your point on the circle to the origin.
One angle of this triangle will always be 90 degrees, one angle will be someone you will figure out by using the unit circle, and the other by subtracting the sum of those two values from 180.
From there, it’s a matter of knowing your triangle sides ratios.
Then, finally, remember to know where your triangle is oriented on a coordinate system, so that you assign the proper signs to the values of the sides.
Sorry if this is nothing but a convoluted mess.
tjwoods
Kuvri (yodaluv12)
thegcriti