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Cos 3pi/4
The value of cos 3pi/4 is 0.7071067. . .. Cos 3pi/4 radians in degrees is written as cos ((3π/4) × 180°/π), i.e., cos (135°). In this article, we will discuss the methods to find the value of cos 3pi/4 with examples.
 Cos 3pi/4: −(1/√2)
 Cos 3pi/4 in decimal: 0.7071067. . .
 Cos (3pi/4): 0.7071067. . . or (1/√2)
 Cos 3pi/4 in degrees: cos (135°)
What is the Value of Cos 3pi/4?
The value of cos 3pi/4 in decimal is 0.707106781. . .. Cos 3pi/4 can also be expressed using the equivalent of the given angle (3pi/4) in degrees (135°).
We know, using radian to degree conversion, θ in degrees = θ in radians × (180°/pi)
⇒ 3pi/4 radians = 3pi/4 × (180°/pi) = 135° or 135 degrees
∴ cos 3pi/4 = cos 3π/4 = cos(135°) = −(1/√2) or 0.7071067. . .
Explanation:
For cos 3pi/4, the angle 3pi/4 lies between pi/2 and pi (Second Quadrant). Since cosine function is negative in the second quadrant, thus cos 3pi/4 value = −(1/√2) or 0.7071067. . .
Since the cosine function is a periodic function, we can represent cos 3pi/4 as, cos 3pi/4 = cos(3pi/4 + n × 2pi), n ∈ Z.
⇒ cos 3pi/4 = cos 11pi/4 = cos 19pi/4 , and so on.
Note: Since, cosine is an even function, the value of cos(3pi/4) = cos(3pi/4).
Methods to Find Value of Cos 3pi/4
The cosine function is negative in the 2nd quadrant. The value of cos 3pi/4 is given as 0.70710. . .. We can find the value of cos 3pi/4 by:
 Using Trigonometric Functions
 Using Unit Circle
Cos 3pi/4 in Terms of Trigonometric Functions
Using trigonometry formulas, we can represent the cos 3pi/4 as:
 ± √(1sin²(3pi/4))
 ± 1/√(1 + tan²(3pi/4))
 ± cot(3pi/4)/√(1 + cot²(3pi/4))
 ±√(cosec²(3pi/4)  1)/cosec(3pi/4)
 1/sec(3pi/4)
Note: Since 3pi/4 lies in the 2nd Quadrant, the final value of cos 3pi/4 will be negative.
We can use trigonometric identities to represent cos 3pi/4 as,
 cos(pi  3pi/4) = cos pi/4
 cos(pi + 3pi/4) = cos 7pi/4
 sin(pi/2 + 3pi/4) = sin 5pi/4
 sin(pi/2  3pi/4) = sin(pi/4)
Cos 3pi/4 Using Unit Circle
To find the value of cos 3π/4 using the unit circle:
 Rotate ‘r’ anticlockwise to form 3pi/4 angle with the positive xaxis.
 The cos of 3pi/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 3pi/4 = x = 0.7071 (approx)
☛ Also Check:
Examples Using Cos 3pi/4

Example 1: Find the value of 2 cos(3pi/4)/3 sin(pi/4).
Solution:
Using trigonometric identities, we know, cos(3pi/4) = sin(pi/2  3pi/4) = sin(pi/4).
⇒ cos(3pi/4) = sin(pi/4)
⇒ Value of 2 cos(3pi/4)/3 sin(pi/4) = 2/3 
Example 2: Find the value of cos 3pi/4 if sec 3pi/4 is 1.4142.
Solution:
Since, cos 3pi/4 = 1/sec(3pi/4)
⇒ cos 3pi/4 = 1/(1.4142) = 0.7071 
Example 3: Find the value of (cos² 3pi/8  sin² 3pi/8). [Hint: Use cos 3pi/4 = 0.7071]
Solution:
Using the cos 2a formula,
(cos² 3pi/8  sin² 3pi/8) = cos(2 × 3pi/8) = cos 3pi/4
∵ cos 3pi/4 = 0.7071
⇒ (cos² 3pi/8  sin² 3pi/8) = 0.7071
FAQs on Cos 3pi/4
What is Cos 3pi/4?
Cos 3pi/4 is the value of cosine trigonometric function for an angle equal to 3π/4 radians. The value of cos 3pi/4 is −(1/√2) or 0.7071 (approx)
What is the Value of Cos 3pi/4 in Terms of Cosec 3pi/4?
Since the cosine function can be represented using the cosecant function, we can write cos 3pi/4 as [√(cosec²(3pi/4)  1)/cosec 3pi/4]. The value of cosec 3pi/4 is equal to 1.41421.
What is the Value of Cos 3pi/4 in Terms of Tan 3pi/4?
We know, using trig identities, we can write cos 3pi/4 as 1/√(1 + tan²(3pi/4)). Here, the value of tan 3pi/4 is equal to 1.
How to Find the Value of Cos 3pi/4?
The value of cos 3pi/4 can be calculated by constructing an angle of 3π/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 3pi/4 is equal to the xcoordinate (0.7071). ∴ cos 3pi/4 = 0.7071.
How to Find Cos 3pi/4 in Terms of Other Trigonometric Functions?
Using trigonometry formula, the value of cos 3pi/4 can be given in terms of other trigonometric functions as:
 ± √(1sin²(3pi/4))
 ± 1/√(1 + tan²(3pi/4))
 ± cot(3pi/4)/√(1 + cot²(3pi/4))
 ±√(cosec²(3pi/4)  1)/cosec(3pi/4)
 1/sec(3pi/4)
☛ Also check: trigonometry table
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