Cos 7pi/3
The value of cos 7pi/3 is 0.5. Cos 7pi/3 radians in degrees is written as cos ((7π/3) × 180°/π), i.e., cos (420°). In this article, we will discuss the methods to find the value of cos 7pi/3 with examples.
- Cos 7pi/3: 1/2
- Cos 7pi/3 in decimal: 0.5
- Cos (-7pi/3): 0.5 or 1/2
- Cos 7pi/3 in degrees: cos (420°)
What is the Value of Cos 7pi/3?
The value of cos 7pi/3 in decimal is 0.5. Cos 7pi/3 can also be expressed using the equivalent of the given angle (7pi/3) in degrees (420°).
We know, using radian to degree conversion, θ in degrees = θ in radians × (180°/pi)
⇒ 7pi/3 radians = 7pi/3 × (180°/pi) = 420° or 420 degrees
∴ cos 7pi/3 = cos 7π/3 = cos(420°) = 1/2 or 0.5
Explanation:
For cos 7pi/3, the angle 7pi/3 > 2pi. Given the periodic property of the cosine function, we can represent it as cos(7pi/3 mod 2pi) = cos(pi/3). The angle 7pi/3, coterminal to angle pi/3, is located in the First Quadrant(Quadrant I).
Since cos function is positive in the 1st quadrant, thus cos 7pi/3 value = 1/2 or 0.5
Similarly, cos 7pi/3 can also be written as, cos 7pi/3 = (7pi/3 + n × 2pi), n ∈ Z.
⇒ cos 7pi/3 = cos 13pi/3 = cos 19pi/3, and so on.
Note: Since, cosine is an even function, the value of cos(-7pi/3) = cos(7pi/3).
Methods to Find Value of Cos 7pi/3
The cosine function is positive in the 1st quadrant. The value of cos 7pi/3 is given as 0.5. We can find the value of cos 7pi/3 by:
- Using Unit Circle
- Using Trigonometric Functions
Cos 7pi/3 Using Unit Circle
To find the value of cos 7π/3 using the unit circle, represent 7pi/3 in the form (1 × 2pi) + pi/3 [∵ 7pi/3>2pi] ∵ cosine is a periodic function, cos 7pi/3 = cos pi/3.
- Rotate ‘r’ anticlockwise to form pi/3 or 7pi/3 angle with the positive x-axis.
- The cos of 7pi/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 7pi/3 = x = 0.5
Cos 7pi/3 in Terms of Trigonometric Functions
Using trigonometry formulas, we can represent the cos 7pi/3 as:
- ± √(1-sin²(7pi/3))
- ± 1/√(1 + tan²(7pi/3))
- ± cot(7pi/3)/√(1 + cot²(7pi/3))
- ±√(cosec²(7pi/3) - 1)/cosec(7pi/3)
- 1/sec(7pi/3)
Note: Since 7pi/3 lies in the 1st Quadrant, the final value of cos 7pi/3 will be positive.
We can use trigonometric identities to represent cos 7pi/3 as,
- -cos(pi - 7pi/3) = -cos(-4pi/3)
- -cos(pi + 7pi/3) = -cos 10pi/3
- sin(pi/2 + 7pi/3) = sin 17pi/6
- sin(pi/2 - 7pi/3) = sin(-11pi/6)
☛ Also Check:
FAQs on Cos 7pi/3
What is Cos 7pi/3?
Cos 7pi/3 is the value of cosine trigonometric function for an angle equal to 7π/3 radians. The value of cos 7pi/3 is 1/2 or 0.5
What is the Value of Cos 7pi/3 in Terms of Sec 7pi/3?
Since the secant function is the reciprocal of the cosine function, we can write cos 7pi/3 as 1/sec(7pi/3). The value of sec 7pi/3 is equal to 2.
What is the Value of Cos 7pi/3 in Terms of Cot 7pi/3?
We can represent the cosine function in terms of the cotangent function using trig identities, cos 7pi/3 can be written as cot(7pi/3)/√(1 + cot²(7pi/3)). Here, the value of cot 7pi/3 is equal to 0.57735.
How to Find Cos 7pi/3 in Terms of Other Trigonometric Functions?
Using trigonometry formula, the value of cos 7pi/3 can be given in terms of other trigonometric functions as:
- ± √(1-sin²(7pi/3))
- ± 1/√(1 + tan²(7pi/3))
- ± cot(7pi/3)/√(1 + cot²(7pi/3))
- ±√(cosec²(7pi/3) - 1)/cosec(7pi/3)
- 1/sec(7pi/3)
☛ Also check: trigonometry table
How to Find the Value of Cos 7pi/3?
The value of cos 7pi/3 can be calculated by constructing an angle of 7π/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 7pi/3 is equal to the x-coordinate (0.5). ∴ cos 7pi/3 = 0.5.