Cos 10pi/3
The value of cos 10pi/3 is -0.5. Cos 10pi/3 radians in degrees is written as cos ((10π/3) × 180°/π), i.e., cos (600°). In this article, we will discuss the methods to find the value of cos 10pi/3 with examples.
- Cos 10pi/3: -(1/2)
- Cos 10pi/3 in decimal: -0.5
- Cos (-10pi/3): -0.5 or -(1/2)
- Cos 10pi/3 in degrees: cos (600°)
What is the Value of Cos 10pi/3?
The value of cos 10pi/3 in decimal is -0.5. Cos 10pi/3 can also be expressed using the equivalent of the given angle (10pi/3) in degrees (600°).
We know, using radian to degree conversion, θ in degrees = θ in radians × (180°/pi)
⇒ 10pi/3 radians = 10pi/3 × (180°/pi) = 600° or 600 degrees
∴ cos 10pi/3 = cos 10π/3 = cos(600°) = -(1/2) or -0.5
Explanation:
For cos 10pi/3, the angle 10pi/3 > 2pi. Given the periodic property of the cosine function, we can represent it as cos(10pi/3 mod 2pi) = cos(4pi/3). The angle 10pi/3, coterminal to angle 4pi/3, is located in the Third Quadrant(Quadrant III).
Since cos function is negative in the 3rd quadrant, thus cos 10pi/3 value = -(1/2) or -0.5
Similarly, cos 10pi/3 can also be written as, cos 10pi/3 = (10pi/3 + n × 2pi), n ∈ Z.
⇒ cos 10pi/3 = cos 16pi/3 = cos 22pi/3, and so on.
Note: Since, cosine is an even function, the value of cos(-10pi/3) = cos(10pi/3).
Methods to Find Value of Cos 10pi/3
The cosine function is negative in the 3rd quadrant. The value of cos 10pi/3 is given as -0.5. We can find the value of cos 10pi/3 by:
- Using Unit Circle
- Using Trigonometric Functions
Cos 10pi/3 Using Unit Circle
To find the value of cos 10π/3 using the unit circle, represent 10pi/3 in the form (1 × 2pi) + 4pi/3 [∵ 10pi/3>2pi] ∵ cosine is a periodic function, cos 10pi/3 = cos 4pi/3.
- Rotate ‘r’ anticlockwise to form 4pi/3 or 10pi/3 angle with the positive x-axis.
- The cos of 10pi/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 10pi/3 = x = -0.5
Cos 10pi/3 in Terms of Trigonometric Functions
Using trigonometry formulas, we can represent the cos 10pi/3 as:
- ± √(1-sin²(10pi/3))
- ± 1/√(1 + tan²(10pi/3))
- ± cot(10pi/3)/√(1 + cot²(10pi/3))
- ±√(cosec²(10pi/3) - 1)/cosec(10pi/3)
- 1/sec(10pi/3)
Note: Since 10pi/3 lies in the 3rd Quadrant, the final value of cos 10pi/3 will be negative.
We can use trigonometric identities to represent cos 10pi/3 as,
- -cos(pi - 10pi/3) = -cos(-7pi/3)
- -cos(pi + 10pi/3) = -cos 13pi/3
- sin(pi/2 + 10pi/3) = sin 23pi/6
- sin(pi/2 - 10pi/3) = sin(-17pi/6)
☛ Also Check:
FAQs on Cos 10pi/3
What is Cos 10pi/3?
Cos 10pi/3 is the value of cosine trigonometric function for an angle equal to 10π/3 radians. The value of cos 10pi/3 is -(1/2) or -0.5
What is the Value of Cos 10pi/3 in Terms of Tan 10pi/3?
We know, using trig identities, we can write cos 10pi/3 as -1/√(1 + tan²(10pi/3)). Here, the value of tan 10pi/3 is equal to 1.732050.
How to Find Cos 10pi/3 in Terms of Other Trigonometric Functions?
Using trigonometry formula, the value of cos 10pi/3 can be given in terms of other trigonometric functions as:
- ± √(1-sin²(10pi/3))
- ± 1/√(1 + tan²(10pi/3))
- ± cot(10pi/3)/√(1 + cot²(10pi/3))
- ±√(cosec²(10pi/3) - 1)/cosec(10pi/3)
- 1/sec(10pi/3)
☛ Also check: trigonometric table
What is the Value of Cos 10pi/3 in Terms of Cosec 10pi/3?
Since the cosine function can be represented using the cosecant function, we can write cos 10pi/3 as [√(cosec²(10pi/3) - 1)/cosec 10pi/3]. The value of cosec 10pi/3 is equal to -1.15470.
How to Find the Value of Cos 10pi/3?
The value of cos 10pi/3 can be calculated by constructing an angle of 10π/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 10pi/3 is equal to the x-coordinate (-0.5). ∴ cos 10pi/3 = -0.5.