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Sin 405 Degrees
The value of sin 405 degrees is 0.7071067. . .. Sin 405 degrees in radians is written as sin (405° × π/180°), i.e., sin (9π/4) or sin (7.068583. . .). In this article, we will discuss the methods to find the value of sin 405 degrees with examples.
 Sin 405°: 0.7071067. . .
 Sin 405° in fraction: 1/√2
 Sin (405 degrees): 0.7071067. . .
 Sin 405° in radians: sin (9π/4) or sin (7.0685834 . . .)
What is the Value of Sin 405 Degrees?
The value of sin 405 degrees in decimal is 0.707106781. . .. Sin 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 . . .
∴ sin 405° = sin(7.0685) = 1/√2 or 0.7071067. . .
Explanation:
For sin 405°, the angle 405° > 360°. Given the periodic property of the sine function, we can represent it as sin(405° mod 360°) = sin(45°). The angle 405°, coterminal to angle 45°, is located in the First Quadrant(Quadrant I).
Since sine function is positive in the 1st quadrant, thus sin 405 degrees value = 1/√2 or 0.7071067. . .
Similarly, sin 405° can also be written as, sin 405 degrees = (405° + n × 360°), n ∈ Z.
⇒ sin 405° = sin 765° = sin 1125°, and so on.
Note: Since, sine is an odd function, the value of sin(405°) = sin(405°).
Methods to Find Value of Sin 405 Degrees
The sine function is positive in the 1st quadrant. The value of sin 405° is given as 0.70710. . .. We can find the value of sin 405 degrees by:
 Using Unit Circle
 Using Trigonometric Functions
Sin 405 Degrees Using Unit Circle
To find the value of sin 405 degrees using the unit circle, represent 405° in the form (1 × 360°) + 45° [∵ 405°>360°] ∵ sine is a periodic function, sin 405° = sin 45°.
 Rotate ‘r’ anticlockwise to form a 45° or 405° angle with the positive xaxis.
 The sin of 405 degrees equals the ycoordinate(0.7071) of the point of intersection (0.7071, 0.7071) of unit circle and r.
Hence the value of sin 405° = y = 0.7071 (approx)
Sin 405° in Terms of Trigonometric Functions
Using trigonometry formulas, we can represent the sin 405 degrees as:
 ± √(1cos²(405°))
 ± tan 405°/√(1 + tan²(405°))
 ± 1/√(1 + cot²(405°))
 ± √(sec²(405°)  1)/sec 405°
 1/cosec 405°
Note: Since 405° lies in the 1st Quadrant, the final value of sin 405° will be positive.
We can use trigonometric identities to represent sin 405° as,
 sin(180°  405°) = sin(225°)
 sin(180° + 405°) = sin 585°
 cos(90°  405°) = cos(315°)
 cos(90° + 405°) = cos 495°
☛ Also Check:
Examples Using Sin 405 Degrees

Example 1: Find the value of sin 405° if cosec 405° is 1.4142.
Solution:
Since, sin 405° = 1/csc 405°
⇒ sin 405° = 1/1.4142 = 0.7071 
Example 2: Find the value of 5 sin(405°)/7 cos(315°).
Solution:
Using trigonometric identities, we know, sin(405°) = cos(90°  405°) = cos(315°).
⇒ sin(405°) = cos(315°)
⇒ Value of 5 sin(405°)/7 cos(315°) = 5/7 
Example 3: Find the value of 2 × (sin 202.5° cos 202.5°). [Hint: Use sin 405° = 0.7071]
Solution:
Using the sin 2a formula,
2 sin 202.5° cos 202.5° = sin(2 × 202.5°) = sin 405°
∵ sin 405° = 0.7071
⇒ 2 × (sin 202.5° cos 202.5°) = 0.7071
FAQs on Sin 405 Degrees
What is Sin 405 Degrees?
Sin 405 degrees is the value of sine trigonometric function for an angle equal to 405 degrees. The value of sin 405° is 1/√2 or 0.7071 (approx).
How to Find the Value of Sin 405 Degrees?
The value of sin 405 degrees can be calculated by constructing an angle of 405° with the xaxis, and then finding the coordinates of the corresponding point (0.7071, 0.7071) on the unit circle. The value of sin 405° is equal to the ycoordinate (0.7071). ∴ sin 405° = 0.7071.
How to Find Sin 405° in Terms of Other Trigonometric Functions?
Using trigonometry formula, the value of sin 405° can be given in terms of other trigonometric functions as:
 ± √(1cos²(405°))
 ± tan 405°/√(1 + tan²(405°))
 ± 1/√(1 + cot²(405°))
 ± √(sec²(405°)  1)/sec 405°
 1/cosec 405°
☛ Also check: trigonometric table
What is the Value of Sin 405 Degrees in Terms of Tan 405°?
We know, using trig identities, we can write sin 405° as tan 405°/√(1 + tan²(405°)). Here, the value of tan 405° is equal to 1.
What is the Value of Sin 405° in Terms of Cosec 405°?
Since the cosecant function is the reciprocal of the sine function, we can write sin 405° as 1/cosec(405°). The value of cosec 405° is equal to 1.41421.
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