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:

• ± √(1-sin²(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 x-axis.
• The cos of 3pi/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 3pi/4 = x = -0.7071 (approx)

☛ Also Check:

• sin 9pi/2
• sin 11pi/12
• cos 5pi/4
• cos 10pi/3
• cos pi/12
• sin pi/12

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 x-axis, 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 x-coordinate (-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:

• ± √(1-sin²(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