Cos 2pi/3
The value of cos 2pi/3 is -0.5. Cos 2pi/3 radians in degrees is written as cos ((2π/3) × 180°/π), i.e., cos (120°). In this article, we will discuss the methods to find the value of cos 2pi/3 with examples.
- Cos 2pi/3: -(1/2)
- Cos 2pi/3 in decimal: -0.5
- Cos (-2pi/3): -0.5 or -(1/2)
- Cos 2pi/3 in degrees: cos (120°)
What is the Value of Cos 2pi/3?
The value of cos 2pi/3 in decimal is -0.5. Cos 2pi/3 can also be expressed using the equivalent of the given angle (2pi/3) in degrees (120°).
We know, using radian to degree conversion, θ in degrees = θ in radians × (180°/pi)
⇒ 2pi/3 radians = 2pi/3 × (180°/pi) = 120° or 120 degrees
∴ cos 2pi/3 = cos 2π/3 = cos(120°) = -(1/2) or -0.5
Explanation:
For cos 2pi/3, the angle 2pi/3 lies between pi/2 and pi (Second Quadrant). Since cosine function is negative in the second quadrant, thus cos 2pi/3 value = -(1/2) or -0.5
Since the cosine function is a periodic function, we can represent cos 2pi/3 as, cos 2pi/3 = cos(2pi/3 + n × 2pi), n ∈ Z.
⇒ cos 2pi/3 = cos 8pi/3 = cos 14pi/3 , and so on.
Note: Since, cosine is an even function, the value of cos(-2pi/3) = cos(2pi/3).
Methods to Find Value of Cos 2pi/3
The cosine function is negative in the 2nd quadrant. The value of cos 2pi/3 is given as -0.5. We can find the value of cos 2pi/3 by:
- Using Unit Circle
- Using Trigonometric Functions
Cos 2pi/3 Using Unit Circle
To find the value of cos 2π/3 using the unit circle:
- Rotate ‘r’ anticlockwise to form 2pi/3 angle with the positive x-axis.
- The cos of 2pi/3 equals the x-coordinate(-0.5) of the point of intersection (-0.5, 0.866) of unit circle and r.
Hence the value of cos 2pi/3 = x = -0.5
Cos 2pi/3 in Terms of Trigonometric Functions
Using trigonometry formulas, we can represent the cos 2pi/3 as:
- ± √(1-sin²(2pi/3))
- ± 1/√(1 + tan²(2pi/3))
- ± cot(2pi/3)/√(1 + cot²(2pi/3))
- ±√(cosec²(2pi/3) - 1)/cosec(2pi/3)
- 1/sec(2pi/3)
Note: Since 2pi/3 lies in the 2nd Quadrant, the final value of cos 2pi/3 will be negative.
We can use trigonometric identities to represent cos 2pi/3 as,
- -cos(pi - 2pi/3) = -cos pi/3
- -cos(pi + 2pi/3) = -cos 5pi/3
- sin(pi/2 + 2pi/3) = sin 7pi/6
- sin(pi/2 - 2pi/3) = sin(-pi/6)
☛ Also Check:
FAQs on Cos 2pi/3
What is Cos 2pi/3?
Cos 2pi/3 is the value of cosine trigonometric function for an angle equal to 2π/3 radians. The value of cos 2pi/3 is -(1/2) or -0.5
What is the Exact Value of cos 2pi/3?
The exact value of cos 2pi/3 is -0.5.
What is the Value of Cos 2pi/3 in Terms of Tan 2pi/3?
We know, using trig identities, we can write cos 2pi/3 as -1/√(1 + tan²(2pi/3)). Here, the value of tan 2pi/3 is equal to -1.732050.
How to Find the Value of Cos 2pi/3?
The value of cos 2pi/3 can be calculated by constructing an angle of 2π/3 radians with the x-axis, and then finding the coordinates of the corresponding point (-0.5, 0.866) on the unit circle. The value of cos 2pi/3 is equal to the x-coordinate (-0.5). ∴ cos 2pi/3 = -0.5.
How to Find Cos 2pi/3 in Terms of Other Trigonometric Functions?
Using trigonometry formula, the value of cos 2pi/3 can be given in terms of other trigonometric functions as:
- ± √(1-sin²(2pi/3))
- ± 1/√(1 + tan²(2pi/3))
- ± cot(2pi/3)/√(1 + cot²(2pi/3))
- ±√(cosec²(2pi/3) - 1)/cosec(2pi/3)
- 1/sec(2pi/3)
☛ Also check: trigonometry table