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Sec pi/3

The value of sec pi/3 is 2. Sec pi/3 radians in degrees is written as sec ((π/3) × 180°/π), i.e., sec (60°). In this article, we will discuss the methods to find the value of sec pi/3 with examples.

  • Sec pi/3: 2
  • Sec (-pi/3): 2
  • Sec pi/3 in degrees: sec (60°)

What is the Value of Sec pi/3?

The value of sec pi/3 is 2. Sec pi/3 can also be expressed using the equivalent of the given angle (pi/3) in degrees (60°).

We know, using radian to degree conversion, θ in degrees = θ in radians × (180°/pi)
⇒ pi/3 radians = pi/3 × (180°/pi) = 60° or 60 degrees
∴ sec pi/3 = sec π/3 = sec(60°) = 2

Sec pi/3

Explanation:

For sec pi/3, the angle pi/3 lies between 0 and pi/2 (First Quadrant). Since secant function is positive in the first quadrant, thus sec pi/3 value = 2
Since the secant function is a periodic function, we can represent sec pi/3 as, sec pi/3 = sec(pi/3 + n × 2pi), n ∈ Z.
⇒ sec pi/3 = sec 7pi/3 = sec 13pi/3 , and so on.
Note: Since, secant is an even function, the value of sec(-pi/3) = sec(pi/3).

Methods to Find Value of Sec pi/3

The secant function is positive in the 1st quadrant. The value of sec pi/3 is given as 2. We can find the value of sec pi/3 by:

  • Using Trigonometric Functions
  • Using Unit Circle

Sec pi/3 in Terms of Trigonometric Functions

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

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

Note: Since pi/3 lies in the 1st Quadrant, the final value of sec pi/3 will be positive.

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

  • -sec(pi – pi/3) = -sec 2pi/3
  • -sec(pi + pi/3) = -sec 4pi/3
  • cosec(pi/2 + pi/3) = cosec 5pi/6
  • cosec(pi/2 – pi/3) = cosec pi/6

Sec pi/3 Using Unit Circle

value of sec pi/3

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

  • Rotate ‘r’ anticlockwise to form pi/3 angle with the positive x-axis.
  • The sec of pi/3 equals the reciprocal of the x-coordinate(0.5) of the point of intersection (0.5, 0.866) of unit circle and r.

Hence the value of sec pi/3 = 1/x = 2

☛ Also Check:

Examples Using Sec pi/3

  1. Example 1: Find the value of 2 sec(pi/3)/3 cosec(pi/6).

    Solution:

    Using trigonometric identities, we know, sec(pi/3) = cosec(pi/2 – pi/3) = cosec pi/6

    ⇒ sec(pi/3) = cosec(pi/6)
    ⇒ Value of 2 sec(pi/3)/3 cosec(pi/6) = 2/3

  2. Example 2: Find the value of 1/(cos² pi/6 – sin² pi/6). [Hint: Use sec pi/3 = 2]

    Solution:

    Using the cos 2a formula,

    1/(cos² pi/6 – sin² pi/6) = 1/cos(2 × pi/6) = sec pi/3
    ∵ sec pi/3 = 2
    ⇒ 1/(cos² pi/6 – sin² pi/6) = 2

  3. Example 3: Simplify: 8 (sec(pi/3)/cosec(5pi/6))

    Solution:

    We know sec pi/3 = cosec 5pi/6

    ⇒ 8 sec(pi/3)/cosec(5pi/6) = 8 (sec(pi/3)/sec(pi/3))
    = 8(1) = 8

FAQs on Sec pi/3

What is Sec pi/3?

Sec pi/3 is the value of secant trigonometric function for an angle equal to π/3. The value of sec pi/3 is 2.

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

We know, using trig identities, we can write sec pi/3 as √(1 + tan²(pi/3)). Here, the value of tan pi/3 is equal to 1.7321.

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

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

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

☛ Also check: trigonometry table

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

Since the cosine function is the reciprocal of the secant function, we can write sec pi/3 as 1/cos(pi/3). The value of cos pi/3 is equal to 0.5.

How to Find the Value of Sec pi/3?

The value of sec pi/3 can be calculated by constructing an angle of π/3 radians with the x-axis, and then finding the coordinates of the corresponding point (0.5, 0.866) on the unit circle. The value of sec pi/3 is equal to the reciprocal of the x-coordinate(0.5). ∴ sec pi/3 = 2.

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