Sec pi/4
The value of sec pi/4 is 1.4142135. . .. Sec pi/4 radians in degrees is written as sec ((π/4) × 180°/π), i.e., sec (45°). In this article, we will discuss the methods to find the value of sec pi/4 with examples.
 Sec pi/4: √2
 Sec pi/4 in decimal: 1.4142135. . .
 Sec (pi/4): 1.4142135. . . or √2
 Sec pi/4 in degrees: sec (45°)
What is the Value of Sec pi/4?
The value of sec pi/4 in decimal is 1.414213562. . .. Sec pi/4 can also be expressed using the equivalent of the given angle (pi/4) in degrees (45°).
We know, using radian to degree conversion, θ in degrees = θ in radians × (180°/pi)
⇒ pi/4 radians = pi/4 × (180°/pi) = 45° or 45 degrees
∴ sec pi/4 = sec π/4 = sec(45°) = √2 or 1.4142135. . .
Explanation:
For sec pi/4, the angle pi/4 lies between 0 and pi/2 (First Quadrant). Since secant function is positive in the first quadrant, thus sec pi/4 value = √2 or 1.4142135. . .
Since the secant function is a periodic function, we can represent sec pi/4 as, sec pi/4 = sec(pi/4 + n × 2pi), n ∈ Z.
⇒ sec pi/4 = sec 9pi/4 = sec 17pi/4 , and so on.
Note: Since, secant is an even function, the value of sec(pi/4) = sec(pi/4).
Methods to Find Value of Sec pi/4
The secant function is positive in the 1st quadrant. The value of sec pi/4 is given as 1.41421. . .. We can find the value of sec pi/4 by:
 Using Trigonometric Functions
 Using Unit Circle
Sec pi/4 in Terms of Trigonometric Functions
Using trigonometry formulas, we can represent the sec pi/4 as:
 ± 1/√(1  sin²(pi/4))
 ± √(1 + tan²(pi/4))
 ± √(1 + cot²(pi/4))/cot(pi/4)
 ± cosec(pi/4)/√(cosec²(pi/4)  1)
 1/cos(pi/4)
Note: Since pi/4 lies in the 1st Quadrant, the final value of sec pi/4 will be positive.
We can use trigonometric identities to represent sec pi/4 as,
 sec(pi  pi/4) = sec 3pi/4
 sec(pi + pi/4) = sec 5pi/4
 cosec(pi/2 + pi/4) = cosec 3pi/4
 cosec(pi/2  pi/4) = cosec pi/4
Sec pi/4 Using Unit Circle
To find the value of sec π/4 using the unit circle:
 Rotate ‘r’ anticlockwise to form pi/4 angle with the positive xaxis.
 The sec of pi/4 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 pi/4 = 1/x = 1.4142 (approx)
☛ Also Check:
Examples Using Sec pi/4

Example 1: Find the value of 2 sec(pi/4)/3 cosec(pi/4).
Solution:
Using trigonometric identities, we know, sec(pi/4) = cosec(pi/2  pi/4) = cosec pi/4
⇒ sec(pi/4) = cosec(pi/4)
⇒ Value of 2 sec(pi/4)/3 cosec(pi/4) = 2/3 
Example 2: Find the value of 1/(cos² pi/8  sin² pi/8). [Hint: Use sec pi/4 = 1.4142]
Solution:
Using the cos 2a formula,
1/(cos² pi/8  sin² pi/8) = 1/cos(2 × pi/8) = sec pi/4
∵ sec pi/4 = 1.4142
⇒ 1/(cos² pi/8  sin² pi/8) = 1.4142 
Example 3: Find the value of sec pi/4 if cos pi/4 is 0.7071.
Solution:
Since, sec pi/4 = 1/cos(pi/4)
⇒ sec pi/4 = 1/0.7071 = 1.4142
FAQs on Sec pi/4
What is Sec pi/4?
Sec pi/4 is the value of secant trigonometric function for an angle equal to π/4. The value of sec pi/4 is √2 or 1.4142 (approx).
What is the Value of Sec pi/4 in Terms of Tan pi/4?
We know, using trig identities, we can write sec pi/4 as √(1 + tan²(pi/4)). Here, the value of tan pi/4 is equal to 1.
How to Find Sec pi/4 in Terms of Other Trigonometric Functions?
Using trigonometry formula, the value of sec pi/4 can be given in terms of other trigonometric functions as:
 ± 1/√(1  sin²(pi/4))
 ± √(1 + tan²(pi/4))
 ± √(1 + cot²(pi/4))/cot(pi/4)
 ± cosec(pi/4)/√(cosec²(pi/4)  1)
 1/cos(pi/4)
☛ Also check: trigonometric table
What is the Exact Value of Sec pi/4?
The exact value of sec pi/4 can be given accurately up to 8 decimal places as 1.41421356 or as √2.
How to Find the Value of Sec pi/4?
The value of sec pi/4 can be calculated by constructing an angle of π/4 radians with the xaxis, and then finding the coordinates of the corresponding point (0.7071, 0.7071) on the unit circle. The value of sec pi/4 is equal to the reciprocal of the xcoordinate(0.7071). ∴ sec pi/4 = 1.4142.
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