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Sec 390 Degrees

The value of Sec 390 degrees is 1.1547005. . .. Sec 390 degrees in radians is written as sec (390° × π/180°), i.e., sec (13π/6) or sec (6.806784. . .). In this article, we will discuss the methods to find the value of sec 390 degrees with examples.

Sec 390°: 2/√3
Sec 390° in decimal: 1.1547005. . .
Sec (-390 degrees): 1.1547005. . . or 2/√3
Sec 390° in radians: sec (13π/6) or sec (6.8067840 . . .)

What is the Value of Sec 390 Degrees?

The value of sec 390 degrees in decimal is 1.154700538. . .. Sec 390 degrees can also be expressed using the equivalent of the given angle (390 degrees) in radians (6.80678 . . .)

We know, using degree to radian conversion, θ in radians = θ in degrees × (pi/180°)
⇒ 390 degrees = 390° × (π/180°) rad = 13π/6 or 6.8067 . . .
∴ sec 390° = sec(6.8067) = 2/√3 or 1.1547005. . .

Explanation:

For sec 390°, the angle 390° > 360°. Given the periodic property of the secant function, we can represent it as sec(390° mod 360°) = sec(30°). The angle 390°, coterminal to angle 30°, is located in the First Quadrant(Quadrant I).
Since secant function is positive in the 1st quadrant, thus sec 390 degrees value = 2/√3 or 1.1547005. . .
Similarly, sec 390° can also be written as, sec 390 degrees = (390° + n × 360°), n ∈ Z.
⇒ sec 390° = sec 750° = sec 1110°, and so on.
Note: Since, secant is an even function, the value of sec(-390°) = sec(390°).

Methods to Find Value of Sec 390 Degrees

The secant function is positive in the 1st quadrant. The value of sec 390° is given as 1.15470. . .. We can find the value of sec 390 degrees by:

Using Trigonometric Functions
Using Unit Circle

Sec 390° in Terms of Trigonometric Functions

Using trigonometry formulas, we can represent the sec 390 degrees as:

± 1/√(1 - sin²(390°))
± √(1 + tan²(390°))
± √(1 + cot²(390°))/cot 390°
± cosec 390°/√(cosec²(390°) - 1)
1/cos 390°

Note: Since 390° lies in the 1st Quadrant, the final value of sec 390° will be positive.

We can use trigonometric identities to represent sec 390° as,

-sec(180° - 390°) = -sec(-210°)
-sec(180° + 390°) = -sec 570°
cosec(90° + 390°) = cosec 480°
cosec(90° - 390°) = cosec(-300°)

Sec 390 Degrees Using Unit Circle

To find the value of sec 390 degrees using the unit circle, represent 390° in the form (1 × 360°) + 30° [∵ 390°>360°] ∵ secant is a periodic function, sec 390° = sec 30°.

Rotate ‘r’ anticlockwise to form 30° or 390° angle with the positive x-axis.
The sec of 390 degrees equals the reciprocal of the x-coordinate(0.866) of the point of intersection (0.866, 0.5) of unit circle and r.

Hence the value of sec 390° = 1/x = 1.1547 (approx)

☛ Also Check:

Examples Using Sec 390 Degrees

Example 1: Using the value of sec 390°, solve: (1 + tan²(390°)).

Solution:
We know, (1 + tan²(390°)) = (sec²(390°)) = 1.3333
⇒ (1 + tan²(390°)) = 1.3333
Example 2: Find the value of 1/(cos² 195° - sin² 195°). [Hint: Use sec 390° = 1.1547]

Solution:

Using the cos 2a formula,
1/(cos² 195° - sin² 195°) = 1/cos(2 × 195°) = sec 390°
∵ sec 390° = 1.1547
⇒ 1/(cos² 195° - sin² 195°) = 1.1547
Example 3: Find the value of 2 sec(390°)/3 cosec(-300°).

Solution:

Using trigonometric identities, we know, sec(390°) = cosec(90° - 390°) = cosec(-300°).
⇒ sec(390°) = cosec(-300°)
⇒ Value of 2 sec(390°)/3 cosec(-300°) = 2/3

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FAQs on Sec 390 Degrees

What is Sec 390 Degrees?

Sec 390 degrees is the value of secant trigonometric function for an angle equal to 390 degrees. The value of sec 390° is 2/√3 or 1.1547 (approx).

What is the Value of Sec 390° in Terms of Cos 390°?

Since the cosine function is the reciprocal of the secant function, we can write sec 390° as 1/cos(390°). The value of cos 390° is equal to 0.866.

How to Find the Value of Sec 390 Degrees?

The value of sec 390 degrees can be calculated by constructing an angle of 390° with the x-axis, and then finding the coordinates of the corresponding point (0.866, 0.5) on the unit circle. The value of sec 390° is equal to the reciprocal of the x-coordinate(0.866). ∴ sec 390° = 1.1547.

What is the Value of Sec 390 Degrees in Terms of Sin 390°?

Using trigonometric identities, we can write sec 390° in terms of sin 390° as, sec(390°) = 1/√(1 - sin²(390°)). Here, the value of sin 390° is equal to 0.5.

How to Find Sec 390° in Terms of Other Trigonometric Functions?

Using trigonometry formula, the value of sec 390° can be given in terms of other trigonometric functions as:

± 1/√(1-sin²(390°))
± √(1 + tan²(390°))
± √(1 + cot²(390°))/cot 390°
± cosec 390°/√(cosec²(390°) - 1)
1/cos 390°

☛ Also check: trigonometric table

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