Cos pi/4
The value of cos pi/4 is 0.7071067. . .. Cos pi/4 radians in degrees is written as cos ((π/4) × 180°/π), i.e., cos (45°). In this article, we will discuss the methods to find the value of cos pi/4 with examples.
- Cos pi/4: 1/√2
- Cos pi/4 in decimal: 0.7071067. . .
- Cos (-pi/4): 0.7071067. . . or 1/√2
- Cos pi/4 in degrees: cos (45°)
What is the Value of Cos pi/4?
The value of cos pi/4 in decimal is 0.707106781. . .. Cos pi/4 can also be expressed using the equivalent of the given angle (pi/4) in degrees (45°).
We know, using radian to degree conversion, θ in degrees = θ in radians × (180°/pi)
⇒ pi/4 radians = pi/4 × (180°/pi) = 45° or 45 degrees
∴ cos pi/4 = cos π/4 = cos(45°) = 1/√2 or 0.7071067. . .
Explanation:
For cos pi/4, the angle pi/4 lies between 0 and pi/2 (First Quadrant). Since cosine function is positive in the first quadrant, thus cos pi/4 value = 1/√2 or 0.7071067. . .
Since the cosine function is a periodic function, we can represent cos pi/4 as, cos pi/4 = cos(pi/4 + n × 2pi), n ∈ Z.
⇒ cos pi/4 = cos 9pi/4 = cos 17pi/4 , and so on.
Note: Since, cosine is an even function, the value of cos(-pi/4) = cos(pi/4).
Methods to Find Value of Cos pi/4
The cosine function is positive in the 1st quadrant. The value of cos pi/4 is given as 0.70710. . .. We can find the value of cos pi/4 by:
- Using Trigonometric Functions
- Using Unit Circle
Cos pi/4 in Terms of Trigonometric Functions
Using trigonometry formulas, we can represent the cos pi/4 as:
- ± √(1-sin²(pi/4))
- ± 1/√(1 + tan²(pi/4))
- ± cot(pi/4)/√(1 + cot²(pi/4))
- ±√(cosec²(pi/4) - 1)/cosec(pi/4)
- 1/sec(pi/4)
Note: Since pi/4 lies in the 1st Quadrant, the final value of cos pi/4 will be positive.
We can use trigonometric identities to represent cos pi/4 as,
- -cos(pi - pi/4) = -cos 3pi/4
- -cos(pi + pi/4) = -cos 5pi/4
- sin(pi/2 + pi/4) = sin 3pi/4
- sin(pi/2 - pi/4) = sin pi/4
Cos pi/4 Using Unit Circle
To find the value of cos π/4 using the unit circle:
- Rotate ‘r’ anticlockwise to form pi/4 angle with the positive x-axis.
- The cos of pi/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 pi/4 = x = 0.7071 (approx)
☛ Also Check:
FAQs on Cos pi/4
What is Cos pi/4?
Cos pi/4 is the value of cosine trigonometric function for an angle equal to π/4 radians. The value of cos pi/4 is 1/√2 or 0.7071 (approx)
How to Find Cos pi/4 in Terms of Other Trigonometric Functions?
Using trigonometry formula, the value of cos pi/4 can be given in terms of other trigonometric functions as:
- ± √(1-sin²(pi/4))
- ± 1/√(1 + tan²(pi/4))
- ± cot(pi/4)/√(1 + cot²(pi/4))
- ±√(cosec²(pi/4) - 1)/cosec(pi/4)
- 1/sec(pi/4)
☛ Also check: trigonometric table
What is the Value of Cos pi/4 in Terms of Cosec pi/4?
Since the cosine function can be represented using the cosecant function, we can write cos pi/4 as [√(cosec²(pi/4) - 1)/cosec pi/4]. The value of cosec pi/4 is equal to 1.41421.
What is the Value of Cos pi/4 in Terms of Tan pi/4?
We know, using trig identities, we can write cos pi/4 as 1/√(1 + tan²(pi/4)). Here, the value of tan pi/4 is equal to 1.
How to Find the Value of Cos pi/4?
The value of cos pi/4 can be calculated by constructing an angle of π/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 pi/4 is equal to the x-coordinate (0.7071). ∴ cos pi/4 = 0.7071.