Question

How do you prove `cos ((2pi)/3)`?

Answer 1

Let's convert to degrees, which are usually easier to work with.

Explanation:

Use the conversion rate `180/pi` to convert to degrees.

`=> (2pi)/3 xx 180/pi`

#= 120 degrees.

We must now determine the reference angle for 120 degrees. Since 120 degrees is in quadrant II, the reference angle is found by using the expression `180 - theta`, where `theta` is the angle in degrees.

Calculating we get a reference angle of 60 degrees. We must now apply our knowledge of the special triangles to continue.

The special triangle that contains 60 degrees is the 30-60-90 degrees, that has side lengths of `1, sqrt(3) and 2`, respectively. So, the hypotenuse measures 2, the side opposite our angle measures `sqrt(3)` and the side adjacent measures `1`.

Applying the definition that cos = adjacent/hypotenuse, we find that our ratio is `1/2`. However, since we're in quadrant II the x axis is negative and therefore our ratio is in fact `-1/2`.

Thus, `cos((2pi)/3) = -1/2`

You can use the acronym `C-A-S-T (Q. 4-3-2-1)` to remember in which quadrants the ratios are positive. For example, we can say with this acronym that cos is positive in quadrant `IV`.

Feel free to ask anything more either on my Socratic dashboard or on the main questions page. I understand that this might at first seem like a long and complicated process.

Practice exercises

1. Find the exact value of each expression.

`a) tan300`

`b) sin((7pi)/6)`

Good luck!