Phoebe Ware

2023-02-25

How to prove $\mathrm{cos}\left(\frac{2\pi}{3}\right)$?

Cailyn Knight

Beginner2023-02-26Added 7 answers

To convert from degrees to a number, use the formula $\frac{180}{\pi}$

$\Rightarrow \frac{2\pi}{3}\times \frac{180}{\pi}$

= 120 degrees.

The reference angle for 120 degrees must now be established. Given that 120 degrees are within quadrant II, the reference angle is determined by the formula $180-\theta$, where $\theta$ denotes the angle's magnitude in degrees.

We arrive at a reference angle of 60 degrees after calculation. Now, in order to proceed, we must use what we know about the special triangles.

The special triangle that contains 60 degrees is the 30-60-90 degrees, that has side lengths of $1,\sqrt{3}\phantom{\rule{1ex}{0ex}}\text{and}\phantom{\rule{1ex}{0ex}}2$, respectively. Thus, the angle opposite ours measures $\sqrt{3}$, the neighboring side measures $1$, and the hypotenuse measures 2.

Our ratio is $\frac{1}{2}$ when we adopt the concept that cos = adjacent/hypotenuse. However, because we are in quadrant II, the x axis is skewed downward, and as a result, our ratio is actually $-\frac{1}{2}$.

Therefore, $\mathrm{cos}\left(\frac{2\pi}{3}\right)=-\frac{1}{2}$

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 $IV$

$\Rightarrow \frac{2\pi}{3}\times \frac{180}{\pi}$

= 120 degrees.

The reference angle for 120 degrees must now be established. Given that 120 degrees are within quadrant II, the reference angle is determined by the formula $180-\theta$, where $\theta$ denotes the angle's magnitude in degrees.

We arrive at a reference angle of 60 degrees after calculation. Now, in order to proceed, we must use what we know about the special triangles.

The special triangle that contains 60 degrees is the 30-60-90 degrees, that has side lengths of $1,\sqrt{3}\phantom{\rule{1ex}{0ex}}\text{and}\phantom{\rule{1ex}{0ex}}2$, respectively. Thus, the angle opposite ours measures $\sqrt{3}$, the neighboring side measures $1$, and the hypotenuse measures 2.

Our ratio is $\frac{1}{2}$ when we adopt the concept that cos = adjacent/hypotenuse. However, because we are in quadrant II, the x axis is skewed downward, and as a result, our ratio is actually $-\frac{1}{2}$.

Therefore, $\mathrm{cos}\left(\frac{2\pi}{3}\right)=-\frac{1}{2}$

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 $IV$