Sec 3pi

The value of sec 3pi is -1. Sec 3pi radians in degrees is written as sec ((3π) × 180°/π), i.e., sec (540°). In this article, we will discuss the methods to find the value of sec 3pi with examples.

• Sec 3pi: -1
• Sec (-3pi): -1
• Sec 3pi in degrees: sec (540°)

What is the Value of Sec 3pi?

The value of sec 3pi is -1. Sec 3pi can also be expressed using the equivalent of the given angle (3pi) in degrees (540°).

We know, using radian to degree conversion, θ in degrees = θ in radians × (180°/pi)
⇒ 3pi radians = 3pi × (180°/pi) = 540° or 540 degrees
∴ sec 3pi = sec 3π = sec(540°) = -1

Explanation:

For sec 3pi, the angle 3pi > 2pi. We can represent sec 3pi as, sec(3pi mod 2pi) = sec(pi). For sec 3pi, the angle 3pi lies on the negative x-axis. Thus, sec 3pi value = -1
Since the secant function is a periodic function, we can represent sec 3pi as, sec 3pi = sec(3pi + n × 2pi), n ∈ Z.
⇒ sec 3pi = sec 5pi = sec 7pi , and so on.
Note: Since, secant is an even function, the value of sec(-3pi) = sec(3pi).

Methods to Find Value of Sec 3pi

The value of sec 3pi is given as -1. We can find the value of sec 3pi by:

• Using Unit Circle
• Using Trigonometric Functions

Sec 3pi Using Unit Circle

To find the value of sec 3π using the unit circle, represent 3pi in the form (1 × 2pi) + pi [∵ 3pi>2pi] ∵ secant is a periodic function, sec 3pi = sec pi.

• Rotate ‘r’ anticlockwise to form pi or 3pi angle with the positive x-axis.
• The sec of 3pi equals the reciprocal of the x-coordinate(-1) of the point of intersection (-1, 0) of unit circle and r.

Hence the value of sec 3pi = 1/x = -1

Sec 3pi in Terms of Trigonometric Functions

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

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

Note: Since 3pi lies on the negative x-axis, the final value of sec 3pi is -1.

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

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

☛ Also Check:

• sin 7pi/6
• cos 17pi/12
• sin 3pi/2
• cot pi/4
• cos 3pi
• cos 5pi/3

FAQs on Sec 3pi

What is Sec 3pi?

Sec 3pi is the value of secant trigonometric function for an angle equal to 3π. The value of sec 3pi is -1.

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

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

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

☛ Also check: trigonometric table

What is the Value of Sec 3pi in Terms of Cosec 3pi?

Since the secant function can be represented using the cosecant function, we can write sec 3pi as -[cosec(3pi)/√(cosec²(3pi) - 1)]. 

How to Find the Value of Sec 3pi?

The value of sec 3pi can be calculated by constructing an angle of 3π radians with the x-axis, and then finding the coordinates of the corresponding point (-1, 0) on the unit circle. The value of sec 3pi is equal to the reciprocal of the x-coordinate(-1). ∴ sec 3pi = -1.

What is the Value of Sec 3pi in Terms of Cot 3pi?

We can represent the secant function in terms of the cotangent function using trig identities, sec 3pi can be written as -√(1 + cot²(3pi))/cot 3pi.