Cos 405 Degrees

The value of cos 405 degrees is 0.7071067. . .. Cos 405 degrees in radians is written as cos (405° × π/180°), i.e., cos (9π/4) or cos (7.068583. . .). In this article, we will discuss the methods to find the value of cos 405 degrees with examples.

• Cos 405°: 0.7071067. . .
• Cos 405° in fraction: 1/√2
• Cos (-405 degrees): 0.7071067. . .
• Cos 405° in radians: cos (9π/4) or cos (7.0685834 . . .)

What is the Value of Cos 405 Degrees?

The value of cos 405 degrees in decimal is 0.707106781. . .. Cos 405 degrees can also be expressed using the equivalent of the given angle (405 degrees) in radians (7.06858 . . .)

We know, using degree to radian conversion, θ in radians = θ in degrees × (pi/180°)
⇒ 405 degrees = 405° × (π/180°) rad = 9π/4 or 7.0685 . . .
∴ cos 405° = cos(7.0685) = 1/√2 or 0.7071067. . .

Explanation:

For cos 405°, the angle 405° > 360°. Given the periodic property of the cosine function, we can represent it as cos(405° mod 360°) = cos(45°). The angle 405°, coterminal to angle 45°, is located in the First Quadrant(Quadrant I).
Since cosine function is positive in the 1st quadrant, thus cos 405 degrees value = 1/√2 or 0.7071067. . .
Similarly, cos 405° can also be written as, cos 405 degrees = (405° + n × 360°), n ∈ Z.
⇒ cos 405° = cos 765° = cos 1125°, and so on.
Note: Since, cosine is an even function, the value of cos(-405°) = cos(405°).

Methods to Find Value of Cos 405 Degrees

The cosine function is positive in the 1st quadrant. The value of cos 405° is given as 0.70710. . .. We can find the value of cos 405 degrees by:

• Using Unit Circle
• Using Trigonometric Functions

Cos 405 Degrees Using Unit Circle

To find the value of cos 405 degrees using the unit circle, represent 405° in the form (1 × 360°) + 45° [∵ 405°>360°] ∵ cosine is a periodic function, cos 405° = cos 45°.

• Rotate ‘r’ anticlockwise to form 45° or 405° angle with the positive x-axis.
• The cos of 405 degrees 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 405° = x = 0.7071 (approx)

Cos 405° in Terms of Trigonometric Functions

Using trigonometry formulas, we can represent the cos 405 degrees as:

• ± √(1-sin²(405°))
• ± 1/√(1 + tan²(405°))
• ± cot 405°/√(1 + cot²(405°))
• ±√(cosec²(405°) - 1)/cosec 405°
• 1/sec 405°

Note: Since 405° lies in the 1st Quadrant, the final value of cos 405° will be positive.

We can use trigonometric identities to represent cos 405° as,

• -cos(180° - 405°) = -cos(-225°)
• -cos(180° + 405°) = -cos 585°
• sin(90° + 405°) = sin 495°
• sin(90° - 405°) = sin(-315°)

☛ Also Check:

• cos 2 degrees
• cos 75 degrees
• cos 225 degrees
• cos 22 degrees
• cos 5 degrees
• cos 127 degrees

FAQs on Cos 405 Degrees

What is Cos 405 Degrees?

Cos 405 degrees is the value of cosine trigonometric function for an angle equal to 405 degrees. The value of cos 405° is 1/√2 or 0.7071 (approx)

How to Find the Value of Cos 405 Degrees?

The value of cos 405 degrees can be calculated by constructing an angle of 405° 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 405° is equal to the x-coordinate (0.7071). ∴ cos 405° = 0.7071.

What is the Value of Cos 405 Degrees in Terms of Cot 405°?

We can represent the cosine function in terms of the cotangent function using trig identities, cos 405° can be written as cot 405°/√(1 + cot²(405°)). Here, the value of cot 405° is equal to 1.

How to Find Cos 405° in Terms of Other Trigonometric Functions?

Using trigonometry formula, the value of cos 405° can be given in terms of other trigonometric functions as:

• ± √(1-sin²(405°))
• ± 1/√(1 + tan²(405°))
• ± cot 405°/√(1 + cot²(405°))
• ± √(cosec²(405°) - 1)/cosec 405°
• 1/sec 405°

☛ Also check: trigonometry table

What is the Exact Value of cos 405 Degrees?

The exact value of cos 405 degrees can be given accurately up to 8 decimal places as 0.70710678 and 1/√2 in fraction.