Cos 5pi/4
The value of cos 5pi/4 is 0.7071067. . .. Cos 5pi/4 radians in degrees is written as cos ((5π/4) × 180°/π), i.e., cos (225°). In this article, we will discuss the methods to find the value of cos 5pi/4 with examples.
 Cos 5pi/4: (1/√2)
 Cos 5pi/4 in decimal: 0.7071067. . .
 Cos (5pi/4): 0.7071067. . . or −(1/√2)
 Cos 5pi/4 in degrees: cos (225°)
What is the Value of Cos 5pi/4?
The value of cos 5pi/4 in decimal is 0.707106781. . .. Cos 5pi/4 can also be expressed using the equivalent of the given angle (5pi/4) in degrees (225°).
We know, using radian to degree conversion, θ in degrees = θ in radians × (180°/pi)
⇒ 5pi/4 radians = 5pi/4 × (180°/pi) = 225° or 225 degrees
∴ cos 5pi/4 = cos 5π/4 = cos(225°) = (1/√2) or 0.7071067. . .
Explanation:
For cos 5pi/4, the angle 5pi/4 lies between pi and 3pi/2 (Third Quadrant). Since cosine function is negative in the third quadrant, thus cos 5pi/4 value = (1/√2) or 0.7071067. . .
Since the cosine function is a periodic function, we can represent cos 5pi/4 as, cos 5pi/4 = cos(5pi/4 + n × 2pi), n ∈ Z.
⇒ cos 5pi/4 = cos 13pi/4 = cos 21pi/4 , and so on.
Note: Since, cosine is an even function, the value of cos(5pi/4) = cos(5pi/4).
Methods to Find Value of Cos 5pi/4
The cosine function is negative in the 3rd quadrant. The value of cos 5pi/4 is given as 0.70710. . .. We can find the value of cos 5pi/4 by:
 Using Trigonometric Functions
 Using Unit Circle
Cos 5pi/4 in Terms of Trigonometric Functions
Using trigonometry formulas, we can represent the cos 5pi/4 as:
 ± √(1sin²(5pi/4))
 ± 1/√(1 + tan²(5pi/4))
 ± cot(5pi/4)/√(1 + cot²(5pi/4))
 ±√(cosec²(5pi/4)  1)/cosec(5pi/4)
 1/sec(5pi/4)
Note: Since 5pi/4 lies in the 3rd Quadrant, the final value of cos 5pi/4 will be negative.
We can use trigonometric identities to represent cos 5pi/4 as,
 cos(pi  5pi/4) = cos(pi/4)
 cos(pi + 5pi/4) = cos 9pi/4
 sin(pi/2 + 5pi/4) = sin 7pi/4
 sin(pi/2  5pi/4) = sin(3pi/4)
Cos 5pi/4 Using Unit Circle
To find the value of cos 5π/4 using the unit circle:
 Rotate ‘r’ anticlockwise to form 5pi/4 angle with the positive xaxis.
 The cos of 5pi/4 equals the xcoordinate(0.7071) of the point of intersection (0.7071, 0.7071) of unit circle and r.
Hence the value of cos 5pi/4 = x = 0.7071 (approx)
☛ Also Check:
Examples Using Cos 5pi/4

Example 1: Using the value of cos 5pi/4, solve: (1sin²(5pi/4)).
Solution:
We know, (1sin²(5pi/4)) = (cos²(5pi/4)) = 0.5
⇒ (1sin²(5pi/4)) = 0.5 
Example 2: Find the value of cos 5pi/4 if sec 5pi/4 is 1.4142.
Solution:
Since, cos 5pi/4 = 1/sec(5pi/4)
⇒ cos 5pi/4 = 1/(1.4142) = 0.7071 
Example 3: Find the value of (cos² 5pi/8  sin² 5pi/8). [Hint: Use cos 5pi/4 = 0.7071]
Solution:
Using the cos 2a formula,
(cos² 5pi/8  sin² 5pi/8) = cos(2 × 5pi/8) = cos 5pi/4
∵ cos 5pi/4 = 0.7071
⇒ (cos² 5pi/8  sin² 5pi/8) = 0.7071
FAQs on Cos 5pi/4
What is Cos 5pi/4?
Cos 5pi/4 is the value of cosine trigonometric function for an angle equal to 5π/4 radians. The value of cos 5pi/4 is (1/√2) or 0.7071 (approx)
What is the Value of Cos 5pi/4 in Terms of Sec 5pi/4?
Since the secant function is the reciprocal of the cosine function, we can write cos 5pi/4 as 1/sec(5pi/4). The value of sec 5pi/4 is equal to 1.414213.
How to Find the Value of Cos 5pi/4?
The value of cos 5pi/4 can be calculated by constructing an angle of 5π/4 radians with the xaxis, and then finding the coordinates of the corresponding point (0.7071, 0.7071) on the unit circle. The value of cos 5pi/4 is equal to the xcoordinate (0.7071). ∴ cos 5pi/4 = 0.7071.
How to Find Cos 5pi/4 in Terms of Other Trigonometric Functions?
Using trigonometry formula, the value of cos 5pi/4 can be given in terms of other trigonometric functions as:
 ± √(1sin²(5pi/4))
 ± 1/√(1 + tan²(5pi/4))
 ± cot(5pi/4)/√(1 + cot²(5pi/4))
 ±√(cosec²(5pi/4)  1)/cosec(5pi/4)
 1/sec(5pi/4)
☛ Also check: trigonometry table
What is the Value of Cos 5pi/4 in Terms of Sin 5pi/4?
Using trigonometric identities, we can write cos 5pi/4 in terms of sin 5pi/4 as, cos(5pi/4) = √(1  sin²(5pi/4)). Here, the value of sin 5pi/4 is equal to (1/√2).
visual curriculum