Sec 3pi/4

The value of sec 3pi/4 is -1.4142135. . .. Sec 3pi/4 radians in degrees is written as sec ((3π/4) × 180°/π), i.e., sec (135°). In this article, we will discuss the methods to find the value of sec 3pi/4 with examples.

• Sec 3pi/4: -√2
• Sec 3pi/4 in decimal: -1.4142135. . .
• Sec (-3pi/4): -1.4142135. . . or -√2
• Sec 3pi/4 in degrees: sec (135°)

What is the Value of Sec 3pi/4?

The value of sec 3pi/4 in decimal is -1.414213562. . .. Sec 3pi/4 can also be expressed using the equivalent of the given angle (3pi/4) in degrees (135°).

We know, using radian to degree conversion, θ in degrees = θ in radians × (180°/pi)
⇒ 3pi/4 radians = 3pi/4 × (180°/pi) = 135° or 135 degrees
∴ sec 3pi/4 = sec 3π/4 = sec(135°) = -√2 or -1.4142135. . .

Explanation:

For sec 3pi/4, the angle 3pi/4 lies between pi/2 and pi (Second Quadrant). Since secant function is negative in the second quadrant, thus sec 3pi/4 value = -√2 or -1.4142135. . .
Since the secant function is a periodic function, we can represent sec 3pi/4 as, sec 3pi/4 = sec(3pi/4 + n × 2pi), n ∈ Z.
⇒ sec 3pi/4 = sec 11pi/4 = sec 19pi/4 , and so on.
Note: Since, secant is an even function, the value of sec(-3pi/4) = sec(3pi/4).

Methods to Find Value of Sec 3pi/4

The secant function is negative in the 2nd quadrant. The value of sec 3pi/4 is given as -1.41421. . .. We can find the value of sec 3pi/4 by:

• Using Unit Circle
• Using Trigonometric Functions

Sec 3pi/4 Using Unit Circle

To find the value of sec 3π/4 using the unit circle:

• Rotate ‘r’ anticlockwise to form 3pi/4 angle with the positive x-axis.
• The sec of 3pi/4 equals the reciprocal of the x-coordinate(-0.7071) of the point of intersection (-0.7071, 0.7071) of unit circle and r.

Hence the value of sec 3pi/4 = 1/x = -1.4142 (approx)

Sec 3pi/4 in Terms of Trigonometric Functions

Using trigonometry formulas, we can represent the sec 3pi/4 as:

• ± 1/√(1 - sin²(3pi/4))
• ± √(1 + tan²(3pi/4))
• ± √(1 + cot²(3pi/4))/cot(3pi/4)
• ± cosec(3pi/4)/√(cosec²(3pi/4) - 1)
• 1/cos(3pi/4)

Note: Since 3pi/4 lies in the 2nd Quadrant, the final value of sec 3pi/4 will be negative.

We can use trigonometric identities to represent sec 3pi/4 as,

• -sec(pi - 3pi/4) = -sec pi/4
• -sec(pi + 3pi/4) = -sec 7pi/4
• cosec(pi/2 + 3pi/4) = cosec 5pi/4
• cosec(pi/2 - 3pi/4) = cosec(-pi/4)

☛ Also Check:

• cos 2pi/8
• cos 17pi/12
• cos 3pi
• cos 3pi/2
• sec pi/2
• cos 10pi/3

FAQs on Sec 3pi/4

What is Sec 3pi/4?

Sec 3pi/4 is the value of secant trigonometric function for an angle equal to 3π/4. The value of sec 3pi/4 is -√2 or -1.4142 (approx).

What is the Value of Sec 3pi/4 in Terms of Tan 3pi/4?

We know, using trig identities, we can write sec 3pi/4 as -√(1 + tan²(3pi/4)). Here, the value of tan 3pi/4 is equal to -1.

What is the Value of Sec 3pi/4 in Terms of Cos 3pi/4?

Since the cosine function is the reciprocal of the secant function, we can write sec 3pi/4 as 1/cos(3pi/4). The value of cos 3pi/4 is equal to -0.7071.

How to Find the Value of Sec 3pi/4?

The value of sec 3pi/4 can be calculated by constructing an angle of 3π/4 radians with the x-axis, and then finding the coordinates of the corresponding point (-0.7071, 0.7071) on the unit circle. The value of sec 3pi/4 is equal to the reciprocal of the x-coordinate(-0.7071). ∴ sec 3pi/4 = -1.4142.

How to Find Sec 3pi/4 in Terms of Other Trigonometric Functions?

Using trigonometry formula, the value of sec 3pi/4 can be given in terms of other trigonometric functions as:

• ± 1/√(1 - sin²(3pi/4))
• ± √(1 + tan²(3pi/4))
• ± √(1 + cot²(3pi/4))/cot(3pi/4)
• ± cosec(3pi/4)/√(cosec²(3pi/4) - 1)
• 1/cos(3pi/4)

☛ Also check: trigonometric table