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Cos 180 Degrees

The value of cos 180 degrees is -1. Cos 180 degrees in radians is written as cos (180° × π/180°), i.e., cos (π) or cos (3.141592. . .). In this article, we will discuss the methods to find the value of cos 180 degrees with examples.

Cos 180°: -1
Cos (-180 degrees): -1
Cos 180° in radians: cos (π) or cos (3.1415926 . . .)

What is the Value of Cos 180 Degrees?

The value of cos 180 degrees is -1. Cos 180 degrees can also be expressed using the equivalent of the given angle (180 degrees) in radians (3.14159 . . .)

We know, using degree to radian conversion, θ in radians = θ in degrees × (pi/180°)
⇒ 180 degrees = 180° × (π/180°) rad = π or 3.1415 . . .
∴ cos 180° = cos(3.1415) = -1

Explanation:

For cos 180 degrees, the angle 180° lies on the negative x-axis. Thus cos 180° value = -1
Since the cosine function is a periodic function, we can represent cos 180° as, cos 180 degrees = cos(180° + n × 360°), n ∈ Z.
⇒ cos 180° = cos 540° = cos 900°, and so on.
Note: Since, cosine is an even function, the value of cos(-180°) = cos(180°).

Methods to Find Value of Cos 180 Degrees

The value of cos 180° is given as -1. We can find the value of cos 180 degrees by:

Using Unit Circle
Using Trigonometric Functions

Cos 180 Degrees Using Unit Circle

To find the value of cos 180 degrees using the unit circle:

Rotate ‘r’ anticlockwise to form 180° angle with the positive x-axis.
The cos of 180 degrees equals the x-coordinate(-1) of the point of intersection (-1, 0) of unit circle and r.

Hence the value of cos 180° = x = -1

Cos 180° in Terms of Trigonometric Functions

Using trigonometry formulas, we can represent the cos 180 degrees as:

± √(1-sin²(180°))
± 1/√(1 + tan²(180°))
± cot 180°/√(1 + cot²(180°))
±√(cosec²(180°) - 1)/cosec 180°
1/sec 180°

Note: Since 180° lies on the negative x-axis, the final value of cos 180° will be negative.

We can use trigonometric identities to represent cos 180° as,

-cos(180° - 180°) = -cos 0°
-cos(180° + 180°) = -cos 360°
sin(90° + 180°) = sin 270°
sin(90° - 180°) = sin(-90°)

☛ Also Check:

Examples Using Cos 180 Degrees

Example 1: Find the value of 2 cos(180°)/3 sin(-90°).

Solution:

Using trigonometric identities, we know, cos(180°) = sin(90° - 180°) = sin(-90°).
⇒ cos(180°) = sin(-90°)
⇒ Value of 2 cos(180°)/3 sin(-90°) = 2/3
Example 2: Using the value of cos 180°, solve: (1-sin²(180°)).

Solution:
We know, (1-sin²(180°)) = (cos²(180°)) = 1
⇒ (1-sin²(180°)) = 1
Example 3: Find the value of (cos² 90° - sin² 90°). [Hint: Use cos 180° = -1]

Solution:

Using the cos 2a formula,
(cos² 90° - sin² 90°) = cos(2 × 90°) = cos 180°
∵ cos 180° = -1
⇒ (cos² 90° - sin² 90°) = -1

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FAQs on Cos 180 Degrees

What is Cos 180 Degrees?

Cos 180 degrees is the value of cosine trigonometric function for an angle equal to 180 degrees. The value of cos 180° is -1

What is the Value of Cos 180 Degrees in Terms of Cot 180°?

We can represent the cosine function in terms of the cotangent function using trig identities, cos 180° can be written as -cot 180°/√(1 + cot²(180°)).

How to Find the Value of Cos 180 Degrees?

The value of cos 180 degrees can be calculated by constructing an angle of 180° with the x-axis, and then finding the coordinates of the corresponding point (-1, 0) on the unit circle. The value of cos 180° is equal to the x-coordinate (-1). ∴ cos 180° = -1.

What is the Exact Value of cos 180 Degrees?

The exact value of cos 180 degrees is -1.

How to Find Cos 180° in Terms of Other Trigonometric Functions?

Using trigonometry formula, the value of cos 180° can be given in terms of other trigonometric functions as:

± √(1-sin²(180°))
± 1/√(1 + tan²(180°))
± cot 180°/√(1 + cot²(180°))
± √(cosec²(180°) - 1)/cosec 180°
1/sec 180°

☛ Also check: trigonometry table

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