Cos 3pi

The value of cos 3pi is -1. Cos 3pi radians in degrees is written as cos ((3π) × 180°/π), i.e., cos (540°). In this article, we will discuss the methods to find the value of cos 3pi with examples.

• Cos 3pi: -1
• Cos (-3pi): -1
• Cos 3pi in degrees: cos (540°)

What is the Value of Cos 3pi?

The value of cos 3pi is -1. Cos 3pi can also be expressed using the equivalent of the given angle (3pi) in degrees (540°).

We know, using radian to degree conversion, θ in degrees = θ in radians × (180°/pi)
⇒ 3pi radians = 3pi × (180°/pi) = 540° or 540 degrees
∴ cos 3pi = cos 3π = cos(540°) = -1

Explanation:

For cos 3pi, the angle 3pi > 2pi. We can represent cos 3pi as, cos(3pi mod 2pi) = cos(pi). For cos 3pi, the angle 3pi lies on the negative x-axis. Thus, cos 3pi value = -1
Since the cosine function is a periodic function, we can represent cos 3pi as, cos 3pi = cos(3pi + n × 2pi), n ∈ Z.
⇒ cos 3pi = cos 5pi = cos 7pi , and so on.
Note: Since, cosine is an even function, the value of cos(-3pi) = cos(3pi).

Methods to Find Value of Cos 3pi

The value of cos 3pi is given as -1. We can find the value of cos 3pi by:

• Using Trigonometric Functions
• Using Unit Circle

Cos 3pi in Terms of Trigonometric Functions

Using trigonometry formulas, we can represent the cos 3pi as:

• ± √(1-sin²(3pi))
• ± 1/√(1 + tan²(3pi))
• ± cot(3pi)/√(1 + cot²(3pi))
• ±√(cosec²(3pi) - 1)/cosec(3pi)
• 1/sec(3pi)

Note: Since 3pi lies on the negative x-axis, the final value of cos 3pi is -1.

We can use trigonometric identities to represent cos 3pi as,

• -cos(pi - 3pi) = -cos(-2pi)
• -cos(pi + 3pi) = -cos 4pi
• sin(pi/2 + 3pi) = sin 7pi/2
• sin(pi/2 - 3pi) = sin(-5pi/2)

Cos 3pi Using Unit Circle

To find the value of cos 3π using the unit circle, represent 3pi in the form (1 × 2pi) + pi [∵ 3pi>2pi] ∵ cosine is a periodic function, cos 3pi = cos pi.

• Rotate ‘r’ anticlockwise to form pi or 3pi angle with the positive x-axis.
• The cos of 3pi equals the x-coordinate(-1) of the point of intersection (-1, 0) of unit circle and r.

Hence the value of cos 3pi = x = -1

☛ Also Check:

• sin 2pi/3
• sin pi/12
• sin pi
• cot 5pi/3
• cot 2pi/3
• cos 15pi/4

FAQs on Cos 3pi

What is Cos 3pi?

Cos 3pi is the value of cosine trigonometric function for an angle equal to 3π radians. The value of cos 3pi is -1.

How to Find the Value of Cos 3pi?

The value of cos 3pi can be calculated by constructing an angle of 3π radians with the x-axis, and then finding the coordinates of the corresponding point (-1, 0) on the unit circle. The value of cos 3pi is equal to the x-coordinate (-1). ∴ cos 3pi = -1.

What is the Value of Cos 3pi in Terms of Cosec 3pi?

Since the cosine function can be represented using the cosecant function, we can write cos 3pi as -[√(cosec²(3pi) - 1)/cosec 3pi].

What is the Value of Cos 3pi in Terms of Cot 3pi?

We can represent the cosine function in terms of the cotangent function using trig identities, cos 3pi can be written as -cot(3pi)/√(1 + cot²(3pi)). 

How to Find Cos 3pi in Terms of Other Trigonometric Functions?

Using trigonometry formula, the value of cos 3pi can be given in terms of other trigonometric functions as:

• ± √(1-sin²(3pi))
• ± 1/√(1 + tan²(3pi))
• ± cot(3pi)/√(1 + cot²(3pi))
• ±√(cosec²(3pi) - 1)/cosec(3pi)
• 1/sec(3pi)

☛ Also check: trigonometry table