Hi there! Before we start with your problem. Recall the Trigonometric Function of (-x) first.
- sin(-x) = -sinx
- cos(-x) = cosx
- tan(-x) = -tanx
Now what we need to do is to imagine a unit circle. We know that angle (-x) rotate clockwise while (x) rotates counterclockwise.
Now imagine if it rotates clockwise, it:d start at the fourth quadrant first. That means the y-value is negative while x-value is positive.
We also know that (x, y) = (cos, sin)
Now move to the 2(pi)/3.
[tex] \large{ \frac{2 \pi}{ 3} = \frac{2(180)}{3} \longrightarrow 2(60)= \boxed{120 \degree}}[/tex]
Thus, the value of 2pi/3 is 120°
Now imagine again. In a unit circle, 120° is in second quadrant if we rotate counterclockwise. But if we rotate clockwise, where would the 120° be? Remember that (-x) rotates clockwise. That's right! (-120°) is in the third quadrant. In the third quadrant, cos is negative there along with sin.
So we can conclude that:
- cos(-120°) is in third quadrant.
- cos and sin are negative in third quadrant.
120° is also a reference angle of 60° because of 180°-60° = 120°
Thus,
[tex] \large{cos( - \frac{2 \pi}{3} )= cos( - 120 \degree) = cos(120 \degree) \longrightarrow - \frac{1}{2} }[/tex]
Answer/Conclusions
- cos(-2pi/3) = -1/2
- cos(-2pi/3) use the reference of 60°
- cos(-2pi/3) is in third quadrant.
