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Sec 180 Degrees

The value of Sec 180 degrees is -1. Sec 180 degrees in radians is written as sec (180° × π/180°), i.e., sec (π) or sec (3.141592. . .). In this article, we will discuss the methods to find the value of sec 180 degrees with examples.

Sec 180°: -1
Sec (-180 degrees): -1
Sec 180° in radians: sec (π) or sec (3.1415926 . . .)

What is the Value of Sec 180 Degrees?

The value of sec 180 degrees is -1. Sec 180 degrees can also be expressed using the equivalent of the given angle (180 degrees) in radians (3.14159 . . .)

We know, using degree to radian conversion, θ in radians = θ in degrees × (pi/180°)
⇒ 180 degrees = 180° × (π/180°) rad = π or 3.1415 . . .
∴ sec 180° = sec(3.1415) = -1

Explanation:

For sec 180 degrees, the angle 180° lies on the negative x-axis. Thus sec 180° value = -1
Since the secant function is a periodic function, we can represent sec 180° as, sec 180 degrees = sec(180° + n × 360°), n ∈ Z.
⇒ sec 180° = sec 540° = sec 900°, and so on.
Note: Since, secant is an even function, the value of sec(-180°) = sec(180°) = -1.

Methods to Find Value of Sec 180 Degrees

The value of sec 180° is given as -1. We can find the value of sec 180 degrees by:

Using Unit Circle
Using Trigonometric Functions

Sec 180 Degrees Using Unit Circle

To find the value of sec 180 degrees using the unit circle:

Rotate ‘r’ anticlockwise to form 180° angle with the positive x-axis.
The sec of 180 degrees 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 180° = 1/x = -1

Sec 180° in Terms of Trigonometric Functions

Using trigonometry formulas, we can represent the sec 180 degrees as:

± 1/√(1 - sin²(180°))
± √(1 + tan²(180°))
± √(1 + cot²(180°))/cot 180°
± cosec 180°/√(cosec²(180°) - 1)
1/cos 180°

We can use trigonometric identities to represent sec 180° as,

-sec(180° - 180°) = -sec 0°
-sec(180° + 180°) = -sec 360°
cosec(90° + 180°) = cosec 270°
cosec(90° - 180°) = cosec(-90°)

Note: Since 180° lies on the negative x-axis, the final value of sec 180° will be negative.

☛ Also Check:

Examples Using Sec 180 Degrees

Example 1: Find the value of 2 sec(180°)/3 cosec(-90°).

Solution:

Using trigonometric identities, we know, sec(180°) = cosec(90° - 180°) = cosec(-90°).
⇒ sec(180°) = cosec(-90°)
⇒ Value of 2 sec(180°)/3 cosec(-90°) = 2/3
Example 2: Find the value of sec 180° if cos 180° is -1.

Solution:
Since, sec 180° = 1/cos 180°
⇒ sec 180° = 1/(-1) = -1
Example 3: Simplify: 3 (sec 180°/cosec 270°)

Solution:

We know sec 180° = cosec 270°
⇒ 3 sec 180°/cosec 270° = 3 (sec 180°/sec 180°)
= 3(1) = 3

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FAQs on Sec 180 Degrees

What is Sec 180 Degrees?

Sec 180 degrees is the value of secant trigonometric function for an angle equal to 180 degrees. The value of sec 180° is -1.

What is the Value of Sec 180 Degrees in Terms of Tan 180°?

We know, using trig identities, we can write sec 180° as -√(1 + tan²(180°)). Here, the value of tan 180° is equal to 0.

What is the Exact Value of Sec 180 Degrees?

The exact value of sec 180 degrees is -1.

How to Find the Value of Sec 180 Degrees?

The value of sec 180 degrees can be calculated by constructing an angle of 180° with the x-axis, and then finding the coordinates of the corresponding point (-1, 0) on the unit circle. The value of sec 180° is equal to the reciprocal of the x-coordinate(-1). ∴ sec 180° = -1.

How to Find Sec 180° in Terms of Other Trigonometric Functions?

Using trigonometry formula, the value of sec 180° can be given in terms of other trigonometric functions as:

± 1/√(1-sin²(180°))
± √(1 + tan²(180°))
± √(1 + cot²(180°))/cot 180°
± cosec 180°/√(cosec²(180°) - 1)
1/cos 180°

☛ Also check: trigonometric table

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