Sec 45 Degrees
The value of Sec 45 degrees is 1.4142135. . .. Sec 45 degrees in radians is written as sec (45° × π/180°), i.e., sec (π/4) or sec (0.785398. . .). In this article, we will discuss the methods to find the value of sec 45 degrees with examples.
 Sec 45°: √2
 Sec 45° in decimal: 1.4142135. . .
 Sec (45 degrees): 1.4142135. . . or √2
 Sec 45° in radians: sec (π/4) or sec (0.7853981 . . .)
What is the Value of Sec 45 Degrees?
The value of sec 45 degrees in decimal is 1.414213562. . .. Sec 45 degrees can also be expressed using the equivalent of the given angle (45 degrees) in radians (0.78539 . . .)
We know, using degree to radian conversion, θ in radians = θ in degrees × (pi/180°)
⇒ 45 degrees = 45° × (π/180°) rad = π/4 or 0.7853 . . .
∴ sec 45° = sec(0.7853) = √2 or 1.4142135. . .
Explanation:
For sec 45 degrees, the angle 45° lies between 0° and 90° (First Quadrant). Since secant function is positive in the first quadrant, thus sec 45° value = √2 or 1.4142135. . .
Since the secant function is a periodic function, we can represent sec 45° as, sec 45 degrees = sec(45° + n × 360°), n ∈ Z.
⇒ sec 45° = sec 405° = sec 765°, and so on.
Note: Since, secant is an even function, the value of sec(45°) = sec(45°).
Methods to Find Value of Sec 45 Degrees
The secant function is positive in the 1st quadrant. The value of sec 45° is given as 1.41421. . .. We can find the value of sec 45 degrees by:
 Using Unit Circle
 Using Trigonometric Functions
Sec 45 Degrees Using Unit Circle
To find the value of sec 45 degrees using the unit circle:
 Rotate ‘r’ anticlockwise to form 45° angle with the positive xaxis.
 The sec of 45 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 45° = 1/x = 1.4142 (approx)
Sec 45° in Terms of Trigonometric Functions
Using trigonometry formulas, we can represent the sec 45 degrees as:
 ± 1/√(1  sin²(45°))
 ± √(1 + tan²(45°))
 ± √(1 + cot²(45°))/cot 45°
 ± cosec 45°/√(cosec²(45°)  1)
 1/cos 45°
Note: Since 45° lies in the 1st Quadrant, the final value of sec 45° will be positive.
We can use trigonometric identities to represent sec 45° as,
 sec(180°  45°) = sec 135°
 sec(180° + 45°) = sec 225°
 cosec(90° + 45°) = cosec 135°
 cosec(90°  45°) = cosec 45°
☛ Also Check:
Examples Using Sec 45 Degrees

Example 1: Find the value of 2 sec(45°)/3 cosec(45°).
Solution:
Using trigonometric identities, we know, sec(45°) = cosec(90°  45°) = cosec 45°.
⇒ sec(45°) = cosec(45°)
⇒ Value of 2 sec(45°)/3 cosec(45°) = 2/3 
Example 2: Find the value of 1/(cos² 22.5°  sin² 22.5°). [Hint: Use sec 45° = 1.4142]
Solution:
Using the cos 2a formula,
1/(cos² 22.5°  sin² 22.5°) = 1/cos(2 × 22.5°) = sec 45°
∵ sec 45° = 1.4142
⇒ 1/(cos² 22.5°  sin² 22.5°) = 1.4142 
Example 3: Simplify: 3 (sec 45°/cosec 135°)
Solution:
We know sec 45° = cosec 135°
⇒ 3 sec 45°/cosec 135° = 3 (sec 45°/sec 45°)
= 3(1) = 3
FAQs on Sec 45 Degrees
What is Sec 45 Degrees?
Sec 45 degrees is the value of secant trigonometric function for an angle equal to 45 degrees. The value of sec 45° is √2 or 1.4142 (approx).
What is the Value of Sec 45 Degrees in Terms of Sin 45°?
Using trigonometric identities, we can write sec 45° in terms of sin 45° as, sec(45°) = 1/√(1  sin²(45°)). Here, the value of sin 45° is equal to 0.7071.
How to Find Sec 45° in Terms of Other Trigonometric Functions?
Using trigonometry formula, the value of sec 45° can be given in terms of other trigonometric functions as:
 ± 1/√(1sin²(45°))
 ± √(1 + tan²(45°))
 ± √(1 + cot²(45°))/cot 45°
 ± cosec 45°/√(cosec²(45°)  1)
 1/cos 45°
☛ Also check: trigonometry table
What is the Exact Value of Sec 45 Degrees?
The exact value of sec 45 degrees can be given accurately up to 8 decimal places as 1.41421356 or as √2.
How to Find the Value of Sec 45 Degrees?
The value of sec 45 degrees can be calculated by constructing an angle of 45° with the xaxis, and then finding the coordinates of the corresponding point (0.7071, 0.7071) on the unit circle. The value of sec 45° is equal to the reciprocal of the xcoordinate(0.7071). ∴ sec 45° = 1.4142.