Cos 5pi/8
The value of cos 5pi/8 is -0.3826834. . .. Cos 5pi/8 radians in degrees is written as cos ((5π/8) × 180°/π), i.e., cos (112.5°). In this article, we will discuss the methods to find the value of cos 5pi/8 with examples.
- Cos 5pi/8 in decimal: -0.3826834. . .
- Cos (-5pi/8): -0.3826834. . .
- Cos 5pi/8 in degrees: cos (112.5°)
What is the Value of Cos 5pi/8?
The value of cos 5pi/8 in decimal is -0.382683432. . .. Cos 5pi/8 can also be expressed using the equivalent of the given angle (5pi/8) in degrees (112.5°).
We know, using radian to degree conversion, θ in degrees = θ in radians × (180°/pi)
⇒ 5pi/8 radians = 5pi/8 × (180°/pi) = 112.5° or 112.5 degrees
∴ cos 5pi/8 = cos 5π/8 = cos(112.5°) = -0.3826834. . .
Explanation:
For cos 5pi/8, the angle 5pi/8 lies between pi/2 and pi (Second Quadrant). Since cosine function is negative in the second quadrant, thus cos 5pi/8 value = -0.3826834. . .
Since the cosine function is a periodic function, we can represent cos 5pi/8 as, cos 5pi/8 = cos(5pi/8 + n × 2pi), n ∈ Z.
⇒ cos 5pi/8 = cos 21pi/8 = cos 37pi/8 , and so on.
Note: Since, cosine is an even function, the value of cos(-5pi/8) = cos(5pi/8).
Methods to Find Value of Cos 5pi/8
The cosine function is negative in the 2nd quadrant. The value of cos 5pi/8 is given as -0.38268. . .. We can find the value of cos 5pi/8 by:
- Using Unit Circle
- Using Trigonometric Functions
Cos 5pi/8 Using Unit Circle
To find the value of cos 5π/8 using the unit circle:
- Rotate ‘r’ anticlockwise to form 5pi/8 angle with the positive x-axis.
- The cos of 5pi/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 5pi/8 = x = -0.3827 (approx)
Cos 5pi/8 in Terms of Trigonometric Functions
Using trigonometry formulas, we can represent the cos 5pi/8 as:
- ± √(1-sin²(5pi/8))
- ± 1/√(1 + tan²(5pi/8))
- ± cot(5pi/8)/√(1 + cot²(5pi/8))
- ±√(cosec²(5pi/8) - 1)/cosec(5pi/8)
- 1/sec(5pi/8)
Note: Since 5pi/8 lies in the 2nd Quadrant, the final value of cos 5pi/8 will be negative.
We can use trigonometric identities to represent cos 5pi/8 as,
- -cos(pi - 5pi/8) = -cos 3pi/8
- -cos(pi + 5pi/8) = -cos 13pi/8
- sin(pi/2 + 5pi/8) = sin 9pi/8
- sin(pi/2 - 5pi/8) = sin(-pi/8)
☛ Also Check:
FAQs on Cos 5pi/8
What is Cos 5pi/8?
Cos 5pi/8 is the value of cosine trigonometric function for an angle equal to 5π/8 radians. The value of cos 5pi/8 is -0.3827 (approx)
How to Find Cos 5pi/8 in Terms of Other Trigonometric Functions?
Using trigonometry formula, the value of cos 5pi/8 can be given in terms of other trigonometric functions as:
- ± √(1-sin²(5pi/8))
- ± 1/√(1 + tan²(5pi/8))
- ± cot(5pi/8)/√(1 + cot²(5pi/8))
- ±√(cosec²(5pi/8) - 1)/cosec(5pi/8)
- 1/sec(5pi/8)
☛ Also check: trigonometry table
What is the Value of Cos 5pi/8 in Terms of Cosec 5pi/8?
Since the cosine function can be represented using the cosecant function, we can write cos 5pi/8 as -[√(cosec²(5pi/8) - 1)/cosec 5pi/8]. The value of cosec 5pi/8 is equal to 1.08239.
How to Find the Value of Cos 5pi/8?
The value of cos 5pi/8 can be calculated by constructing an angle of 5π/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 5pi/8 is equal to the x-coordinate (-0.3827). ∴ cos 5pi/8 = -0.3827.
What is the Value of Cos 5pi/8 in Terms of Sin 5pi/8?
Using trigonometric identities, we can write cos 5pi/8 in terms of sin 5pi/8 as, cos(5pi/8) = -√(1 - sin²(5pi/8)). Here, the value of sin 5pi/8 is equal to 0.9239.