Cos 360: Value of Cos 360 Degrees & Trigonometric Identities

Cos 360° is a trigonometric function that represents the angle in the fourth quadrant. Cos 360° is also referred to as cosine 360 degrees. The value of Cos 360° is 1. The angle 360° is greater than 270° and less than or equal to 360°. The angle of 360° also symbolizes full rotation in a xy-plane. The value of cos lies in the fourth quadrant, i.e. between 270° to 360°, and is always positive. Thus, cos 360° is a positive value. Cos 360° in radians is written as cos (360° × π/180°), i.e. cos (2π) or cos (6.283185. . .).

Read More: NCERT Solutions For Class 11 Mathematics Trigonometric Functions

Key Terms: Cos 360 Degrees, Trigonometric Functions, Trigonometric Identities, Angle, Radians, Cosine, Unit Circle, Theta

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Cos 360 Value

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The value of cos 360° is 1. Cos 360 degrees is expressed using the equivalent of the given angle (360 degrees) in radians (6.28318 . . .).

The degrees to radians conversion is done as:

• θ in radians = θ in degrees × (pi/180°)
• 360 degrees = 360° × (π/180°) rad = 2π or 6.2831 . . .

Therefore,

Explanation

The angle 360° lies on the positive x-axis for cos 360 degrees. Thus the value of cos 360° = 1.

Cosine function is a periodic function, which means that it can be represented as, 

• cos 360 degrees = cos(360° + n × 360°), n ∈ Z.
• cos 360° = cos 720° = cos 1080°, and so on.

Also, as cosine is an even function, the value of cos (-360°) = cos (360°).

Cos 360^o Value

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How to Find Cos 360 Degrees?

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The value of cos 360° is equivalent to 1. But, one must be clear with how cos 360 degrees is assigned the value 1. The value of cos 360 degrees can be calculated through two methods which are: 

• Using Trigonometric Functions
• Using Unit Circle

Both of these methods provide us with the value of cos 360° which is 1.

Read More: Trigonometric Functions Important Questions

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Cos 360 Degrees Using Trigonometric Functions

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We can represent cos 360 degrees using trigonometry formulas which are as follows: 

• ± √(1-sin²(360°))
• ± 1/√(1 + tan²(360°))
• ± cot 360°/√(1 + cot²(360°))
• ±√(cosec²(360°) - 1)/cosec 360°
• 1/sec 360°

360° lies on the positive x-axis, which means that the final value of cos 360° will be positive.

Cos 360 Degrees Identities

The trigonometric identities can be used to represent cos 360° as,

• cos 360° = sin (90°+360°) = sin 450°
• cos 360° = sin (90°-360°) = sin -270°
• -cos 360° = cos (180°+360°) = cos 540°
• -cos 360° = cos (180°-360°) = cos -180°

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Cos 360 Degrees Using Unit Circle

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The value of cos 360 degrees can also be calculated using the unit circle:

• Rotate ‘r’ in an anticlockwise direction to form a 0° or 360° angle with the positive x-axis.
• Cos 360 degrees equals the x-coordinate(1) of the point of intersection (1, 0) of the unit circle and r.

Thus, the value of cos 360° = x = 1.

Cos 360 Degrees Using Unit Circle

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Trigonometry Ratio Table

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The trigonometry ratio table represents the values both in radians and degrees which can be used to solve a lot of problems. 

Cos 360 – Theta

The value of the expression cos 360 – theta, i.e. cos(360° – θ) will be 

cos(360° – θ) = cos(4 × 90° – θ)

90° is multiplied by 4 which is an even number, thus, cos will not change. Since 360° – θ comes in the fourth quadrant, cos is always positive. Therefore, 

Cos 360 + Theta

The value of expression cos 360 + theta i.e. cos(360° + θ) will be as follows

cos(360° + θ) = cos(4 × 90° + θ)

90° is multiplied by 4 which is an even number, so cos will not change. Since, 360° + θ comes in the fourth quadrant, where all trigonometric ratios are positive, thus, cos is also positive.

Thus, it can be said that the value of cos 360 + theta is equal to cos theta.

Also Check:

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Solved Examples on Cos 360 Degrees

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Example 1: What will be the value of 7 (cos 360°/sin 450°)?

Solution: It is known that,

cos 360° = sin 450°

So, 7 cos 360°/sin 450° = 7 (cos 360°/cos 360°)

= 7(1) = 7

Thus, the value of 7 (cos 360°/sin 450°) is 7.

Example 2: What will be the value of (cos² 180° - sin² 180°)?

Solution: We will use the cos 2a formula here,

(cos² 180° - sin² 180°) = cos(2 × 180°) = cos 360°

∵ cos 360° = 1

(cos² 180° - sin² 180°) = 1

Thus, the value of (cos² 180° - sin² 180°) is 1.

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Things to Remember

• Cos 360° is a function that symbolizes the angle in the fourth quadrant.
• Cos 360° is also known as Cosine 360.
• The value of cos 360° is 1.
• The value of cos 360° is always positive. 
• Cos 360° in radians is expressed as cos (2π) or cos (6.2831853 . . .).
• The value of Cos 360° can be calculated using trigonometric functions and the unit circle. 

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