Cos pi/12
The value of cos pi/12 is 0.9659258. . .. Cos pi/12 radians in degrees is written as cos ((π/12) × 180°/π), i.e., cos (15°). In this article, we will discuss the methods to find the value of cos pi/12 with examples.
- Cos pi/12: (√6 + √2)/4
- Cos pi/12 in decimal: 0.9659258. . .
- Cos (-pi/12): 0.9659258. . . or (√6 + √2)/4
- Cos pi/12 in degrees: cos (15°)
What is the Value of Cos pi/12?
The value of cos pi/12 in decimal is 0.965925826. . .. Cos pi/12 can also be expressed using the equivalent of the given angle (pi/12) in degrees (15°).
We know, using radian to degree conversion, θ in degrees = θ in radians × (180°/pi)
⇒ pi/12 radians = pi/12 × (180°/pi) = 15° or 15 degrees
∴ cos pi/12 = cos π/12 = cos(15°) = (√6 + √2)/4 or 0.9659258. . .
Explanation:
For cos pi/12, the angle pi/12 lies between 0 and pi/2 (First Quadrant). Since cosine function is positive in the first quadrant, thus cos pi/12 value = (√6 + √2)/4 or 0.9659258. . .
Since the cosine function is a periodic function, we can represent cos pi/12 as, cos pi/12 = cos(pi/12 + n × 2pi), n ∈ Z.
⇒ cos pi/12 = cos 25pi/12 = cos 49pi/12 , and so on.
Note: Since, cosine is an even function, the value of cos(-pi/12) = cos(pi/12).
Methods to Find Value of Cos pi/12
The cosine function is positive in the 1st quadrant. The value of cos pi/12 is given as 0.96592. . .. We can find the value of cos pi/12 by:
- Using Trigonometric Functions
- Using Unit Circle
Cos pi/12 in Terms of Trigonometric Functions
Using trigonometry formulas, we can represent the cos pi/12 as:
- ± √(1-sin²(pi/12))
- ± 1/√(1 + tan²(pi/12))
- ± cot(pi/12)/√(1 + cot²(pi/12))
- ±√(cosec²(pi/12) - 1)/cosec(pi/12)
- 1/sec(pi/12)
Note: Since pi/12 lies in the 1st Quadrant, the final value of cos pi/12 will be positive.
We can use trigonometric identities to represent cos pi/12 as,
- -cos(pi - pi/12) = -cos 11pi/12
- -cos(pi + pi/12) = -cos 13pi/12
- sin(pi/2 + pi/12) = sin 7pi/12
- sin(pi/2 - pi/12) = sin 5pi/12
Cos pi/12 Using Unit Circle
To find the value of cos π/12 using the unit circle:
- Rotate ‘r’ anticlockwise to form pi/12 angle with the positive x-axis.
- The cos of pi/12 equals the x-coordinate(0.9659) of the point of intersection (0.9659, 0.2588) of unit circle and r.
Hence the value of cos pi/12 = x = 0.9659 (approx)
☛ Also Check:
FAQs on Cos pi/12
What is Cos pi/12?
Cos pi/12 is the value of cosine trigonometric function for an angle equal to π/12 radians. The value of cos pi/12 is (√6 + √2)/4 or 0.9659 (approx)
How to Find the Value of Cos pi/12?
The value of cos pi/12 can be calculated by constructing an angle of π/12 radians with the x-axis, and then finding the coordinates of the corresponding point (0.9659, 0.2588) on the unit circle. The value of cos pi/12 is equal to the x-coordinate (0.9659). ∴ cos pi/12 = 0.9659.
What is the Value of Cos pi/12 in Terms of Cosec pi/12?
Since the cosine function can be represented using the cosecant function, we can write cos pi/12 as [√(cosec²(pi/12) - 1)/cosec pi/12]. The value of cosec pi/12 is equal to 3.86370.
What is the Value of Cos pi/12 in Terms of Sin pi/12?
Using trigonometric identities, we can write cos pi/12 in terms of sin pi/12 as, cos(pi/12) = √(1 - sin²(pi/12)). Here, the value of sin pi/12 is equal to (√6 - √2)/4.
How to Find Cos pi/12 in Terms of Other Trigonometric Functions?
Using trigonometry formula, the value of cos pi/12 can be given in terms of other trigonometric functions as:
- ± √(1-sin²(pi/12))
- ± 1/√(1 + tan²(pi/12))
- ± cot(pi/12)/√(1 + cot²(pi/12))
- ±√(cosec²(pi/12) - 1)/cosec(pi/12)
- 1/sec(pi/12)
☛ Also check: trigonometric table