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:
- ± √(1-sin²(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 x-axis.
- The cos of 5pi/4 equals the x-coordinate(-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:
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 x-axis, 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 x-coordinate (-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:
- ± √(1-sin²(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).