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Sec 405 Degrees
The value of Sec 405 degrees is 1.4142135. . .. Sec 405 degrees in radians is written as sec (405° × π/180°), i.e., sec (9π/4) or sec (7.068583. . .). In this article, we will discuss the methods to find the value of sec 405 degrees with examples.
 Sec 405°: √2
 Sec 405° in decimal: 1.4142135. . .
 Sec (405 degrees): 1.4142135. . . or √2
 Sec 405° in radians: sec (9π/4) or sec (7.0685834 . . .)
What is the Value of Sec 405 Degrees?
The value of sec 405 degrees in decimal is 1.414213562. . .. Sec 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 . . .
∴ sec 405° = sec(7.0685) = √2 or 1.4142135. . .
Explanation:
For sec 405°, the angle 405° > 360°. Given the periodic property of the secant function, we can represent it as sec(405° mod 360°) = sec(45°). The angle 405°, coterminal to angle 45°, is located in the First Quadrant(Quadrant I).
Since secant function is positive in the 1st quadrant, thus sec 405 degrees value = √2 or 1.4142135. . .
Similarly, sec 405° can also be written as, sec 405 degrees = (405° + n × 360°), n ∈ Z.
⇒ sec 405° = sec 765° = sec 1125°, and so on.
Note: Since, secant is an even function, the value of sec(405°) = sec(405°).
Methods to Find Value of Sec 405 Degrees
The secant function is positive in the 1st quadrant. The value of sec 405° is given as 1.41421. . .. We can find the value of sec 405 degrees by:
 Using Unit Circle
 Using Trigonometric Functions
Sec 405 Degrees Using Unit Circle
To find the value of sec 405 degrees using the unit circle, represent 405° in the form (1 × 360°) + 45° [∵ 405°>360°] ∵ secant is a periodic function, sec 405° = sec 45°.
 Rotate ‘r’ anticlockwise to form 45° or 405° angle with the positive xaxis.
 The sec of 405 degrees equals the reciprocal of the xcoordinate(0.7071) of the point of intersection (0.7071, 0.7071) of unit circle and r.
Hence the value of sec 405° = 1/x = 1.4142 (approx)
Sec 405° in Terms of Trigonometric Functions
Using trigonometry formulas, we can represent the sec 405 degrees as:
 ± 1/√(1  sin²(405°))
 ± √(1 + tan²(405°))
 ± √(1 + cot²(405°))/cot 405°
 ± cosec 405°/√(cosec²(405°)  1)
 1/cos 405°
Note: Since 405° lies in the 1st Quadrant, the final value of sec 405° will be positive.
We can use trigonometric identities to represent sec 405° as,
 sec(180°  405°) = sec(225°)
 sec(180° + 405°) = sec 585°
 cosec(90° + 405°) = cosec 495°
 cosec(90°  405°) = cosec(315°)
☛ Also Check:
Examples Using Sec 405 Degrees

Example 1: Using the value of sec 405°, solve: (1 + tan²(405°)).
Solution:
We know, (1 + tan²(405°)) = (sec²(405°)) = 2
⇒ (1 + tan²(405°)) = 2 
Example 2: Find the value of 2 sec(405°)/3 cosec(315°).
Solution:
Using trigonometric identities, we know, sec(405°) = cosec(90°  405°) = cosec(315°).
⇒ sec(405°) = cosec(315°)
⇒ Value of 2 sec(405°)/3 cosec(315°) = 2/3 
Example 3: Simplify: 3 (sec 405°/cosec 495°)
Solution:
We know sec 405° = cosec 495°
⇒ 3 sec 405°/cosec 495° = 3 (sec 405°/sec 405°)
= 3(1) = 3
FAQs on Sec 405 Degrees
What is Sec 405 Degrees?
Sec 405 degrees is the value of secant trigonometric function for an angle equal to 405 degrees. The value of sec 405° is √2 or 1.4142 (approx).
How to Find the Value of Sec 405 Degrees?
The value of sec 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 sec 405° is equal to the reciprocal of the xcoordinate(0.7071). ∴ sec 405° = 1.4142.
What is the Value of Sec 405° in Terms of Cosec 405°?
Since the secant function can be represented using the cosecant function, we can write sec 405° as [cosec 405°/√(cosec²(405°)  1)]. The value of cosec 405° is equal to 1.4142.
How to Find Sec 405° in Terms of Other Trigonometric Functions?
Using trigonometry formula, the value of sec 405° can be given in terms of other trigonometric functions as:
 ± 1/√(1sin²(405°))
 ± √(1 + tan²(405°))
 ± √(1 + cot²(405°))/cot 405°
 ± cosec 405°/√(cosec²(405°)  1)
 1/cos 405°
☛ Also check: trigonometric table
What is the Value of Sec 405 Degrees in Terms of Sin 405°?
Using trigonometric identities, we can write sec 405° in terms of sin 405° as, sec(405°) = 1/√(1  sin²(405°)). Here, the value of sin 405° is equal to 0.7071.
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