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Cos pi

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

Cos pi: -1
Cos (-pi): -1
Cos pi in degrees: cos (180°)

What is the Value of Cos pi?

The value of cos pi is -1. Cos pi can also be expressed using the equivalent of the given angle (pi) in degrees (180°).

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

Explanation:

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

Methods to Find Value of Cos pi

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

Using Trigonometric Functions
Using Unit Circle

Cos pi in Terms of Trigonometric Functions

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

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

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

We can use trigonometric identities to represent cos pi as,

-cos(pi - pi) = -cos 0
-cos(pi + pi) = -cos 2pi
sin(pi/2 + pi) = sin 3pi/2
sin(pi/2 - pi) = sin(-pi/2)

Cos pi Using Unit Circle

To find the value of cos π using the unit circle:

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

Hence the value of cos pi = x = -1

☛ Also Check:

Examples Using Cos pi

Example 1: Simplify: 8 (cos(pi)/sin(3pi/2))

Solution:

We know cos pi = sin 3pi/2
⇒ 8 cos(pi)/sin(3pi/2) = 8 (cos(pi)/cos(pi))
= 8(1) = 8
Example 2: Find the value of 2 cos(pi)/3 sin(-pi/2).

Solution:

Using trigonometric identities, we know, cos(pi) = sin(pi/2 - pi) = sin(-pi/2).
⇒ cos(pi) = sin(-pi/2)
⇒ Value of 2 cos(pi)/3 sin(-pi/2) = 2/3
Example 3: Using the value of cos pi, solve: (1-sin²(pi)).

Solution:
We know, (1-sin²(pi)) = (cos²(pi)) = 1
⇒ (1-sin²(pi)) = 1

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FAQs on Cos pi

What is Cos pi?

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

What is the Value of Cos pi in Terms of Sin pi?

Using trigonometric identities, we can write cos pi in terms of sin pi as, cos(pi) = -√(1 - sin²(pi)). Here, the value of sin pi is equal to 0.

How to Find the Value of Cos pi?

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

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

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

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

☛ Also check: trigonometry table

What is the Value of Cos pi in Terms of Sec pi?

Since the secant function is the reciprocal of the cosine function, we can write cos pi as 1/sec(pi). The value of sec pi is equal to -1.

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