Cos 3pi/8
The value of cos 3pi/8 is 0.3826834. . .. Cos 3pi/8 radians in degrees is written as cos ((3π/8) × 180°/π), i.e., cos (67.5°). In this article, we will discuss the methods to find the value of cos 3pi/8 with examples.
- Cos 3pi/8 in decimal: 0.3826834. . .
- Cos (-3pi/8): 0.3826834. . .
- Cos 3pi/8 in degrees: cos (67.5°)
What is the Value of Cos 3pi/8?
The value of cos 3pi/8 in decimal is 0.382683432. . .. Cos 3pi/8 can also be expressed using the equivalent of the given angle (3pi/8) in degrees (67.5°).
We know, using radian to degree conversion, θ in degrees = θ in radians × (180°/pi)
⇒ 3pi/8 radians = 3pi/8 × (180°/pi) = 67.5° or 67.5 degrees
∴ cos 3pi/8 = cos 3π/8 = cos(67.5°) = 0.3826834. . .
Explanation:
For cos 3pi/8, the angle 3pi/8 lies between 0 and pi/2 (First Quadrant). Since cosine function is positive in the first quadrant, thus cos 3pi/8 value = 0.3826834. . .
Since the cosine function is a periodic function, we can represent cos 3pi/8 as, cos 3pi/8 = cos(3pi/8 + n × 2pi), n ∈ Z.
⇒ cos 3pi/8 = cos 19pi/8 = cos 35pi/8 , and so on.
Note: Since, cosine is an even function, the value of cos(-3pi/8) = cos(3pi/8).
Methods to Find Value of Cos 3pi/8
The cosine function is positive in the 1st quadrant. The value of cos 3pi/8 is given as 0.38268. . .. We can find the value of cos 3pi/8 by:
- Using Unit Circle
- Using Trigonometric Functions
Cos 3pi/8 Using Unit Circle
To find the value of cos 3π/8 using the unit circle:
- Rotate ‘r’ anticlockwise to form 3pi/8 angle with the positive x-axis.
- The cos of 3pi/8 equals the x-coordinate(0.3827) of the point of intersection (0.3827, 0.9239) of unit circle and r.
Hence the value of cos 3pi/8 = x = 0.3827 (approx)
Cos 3pi/8 in Terms of Trigonometric Functions
Using trigonometry formulas, we can represent the cos 3pi/8 as:
- ± √(1-sin²(3pi/8))
- ± 1/√(1 + tan²(3pi/8))
- ± cot(3pi/8)/√(1 + cot²(3pi/8))
- ±√(cosec²(3pi/8) - 1)/cosec(3pi/8)
- 1/sec(3pi/8)
Note: Since 3pi/8 lies in the 1st Quadrant, the final value of cos 3pi/8 will be positive.
We can use trigonometric identities to represent cos 3pi/8 as,
- -cos(pi - 3pi/8) = -cos 5pi/8
- -cos(pi + 3pi/8) = -cos 11pi/8
- sin(pi/2 + 3pi/8) = sin 7pi/8
- sin(pi/2 - 3pi/8) = sin pi/8
☛ Also Check:
FAQs on Cos 3pi/8
What is Cos 3pi/8?
Cos 3pi/8 is the value of cosine trigonometric function for an angle equal to 3π/8 radians. The value of cos 3pi/8 is 0.3827 (approx)
What is the Value of Cos 3pi/8 in Terms of Cosec 3pi/8?
Since the cosine function can be represented using the cosecant function, we can write cos 3pi/8 as [√(cosec²(3pi/8) - 1)/cosec 3pi/8]. The value of cosec 3pi/8 is equal to 1.08239.
How to Find the Value of Cos 3pi/8?
The value of cos 3pi/8 can be calculated by constructing an angle of 3π/8 radians with the x-axis, and then finding the coordinates of the corresponding point (0.3827, 0.9239) on the unit circle. The value of cos 3pi/8 is equal to the x-coordinate (0.3827). ∴ cos 3pi/8 = 0.3827.
How to Find Cos 3pi/8 in Terms of Other Trigonometric Functions?
Using trigonometry formula, the value of cos 3pi/8 can be given in terms of other trigonometric functions as:
- ± √(1-sin²(3pi/8))
- ± 1/√(1 + tan²(3pi/8))
- ± cot(3pi/8)/√(1 + cot²(3pi/8))
- ±√(cosec²(3pi/8) - 1)/cosec(3pi/8)
- 1/sec(3pi/8)
☛ Also check: trigonometry table
What is the Value of Cos 3pi/8 in Terms of Cot 3pi/8?
We can represent the cosine function in terms of the cotangent function using trig identities, cos 3pi/8 can be written as cot(3pi/8)/√(1 + cot²(3pi/8)). Here, the value of cot 3pi/8 is equal to 0.41421.