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Tan 390 Degrees

The value of tan 390 degrees is 0.5773502. . .. Tan 390 degrees in radians is written as tan (390° × π/180°), i.e., tan (13π/6) or tan (6.806784. . .). In this article, we will discuss the methods to find the value of tan 390 degrees with examples.

Tan 390°: 1/√3
Tan 390° in decimal: 0.5773502. . .
Tan (-390 degrees): -0.5773502. . . or -1/√3
Tan 390° in radians: tan (13π/6) or tan (6.8067840 . . .)

What is the Value of Tan 390 Degrees?

The value of tan 390 degrees in decimal is 0.577350269. . .. Tan 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 . . .
∴ tan 390° = tan(6.8067) = 1/√3 or 0.5773502. . .

Explanation:

For tan 390°, the angle 390° > 360°. We can represent tan 390° as, tan(390° mod 360°) = tan(30°). The angle 390°, coterminal to angle 30°, is located in the First Quadrant(Quadrant I).
Since tangent function is positive in the 1st quadrant, thus tan 390 degrees value = 1/√3 or 0.5773502. . .
Similarly, given the periodic property of tan 390°, it can also be written as, tan 390 degrees = (390° + n × 180°), n ∈ Z.
⇒ tan 390° = tan 570° = tan 750°, and so on.
Note: Since, tangent is an odd function, the value of tan(-390°) = -tan(390°).

Methods to Find Value of Tan 390 Degrees

The tangent function is positive in the 1st quadrant. The value of tan 390° is given as 0.57735. . .. We can find the value of tan 390 degrees by:

Using Trigonometric Functions
Using Unit Circle

Tan 390° in Terms of Trigonometric Functions

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

sin(390°)/cos(390°)
± sin 390°/√(1 - sin²(390°))
± √(1 - cos²(390°))/cos 390°
± 1/√(cosec²(390°) - 1)
± √(sec²(390°) - 1)
1/cot 390°

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

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

cot(90° - 390°) = cot(-300°)
-cot(90° + 390°) = -cot 480°
-tan (180° - 390°) = -tan(-210°)

Tan 390 Degrees Using Unit Circle

To find the value of tan 390 degrees using the unit circle, represent 390° in the form (1 × 360°) + 30° [∵ 390°>360°] ∵ The angle 390° is coterminal to 30° angle and also tangent is a periodic function, tan 390° = tan 30°.

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

Hence the value of tan 390° = y/x = 0.5774 (approx).

☛ Also Check:

Examples Using Tan 390 Degrees

Example 1: Find the value of 2 tan 195°/(1 - tan²(195°)). [Hint: Use tan 390° = 0.5774]

Solution:

Using the tan 2a formula,
2 tan 195°/(1 - tan²(195°)) = tan(2 × 195°) = tan 390°
∵ tan 390° = 0.5774
⇒ 2 tan 195°/(1 - tan²(195°)) = 0.5774
Example 2: Simplify: 6 (tan 390°/cot(-300°))

Solution:

We know tan 390° = cot(-300°)
⇒ 6 tan 390°/cot(-300°) = 6 (tan 390°/tan 390°)
= 6(1) = 6
Example 3: Using the value of tan 390°, solve: (sec²(390°) - 1).

Solution:
We know, (sec²(390°) - 1) = (tan²(390°)) = 0.3333
⇒ (sec²(390°) - 1) = 0.3333

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FAQs on Tan 390 Degrees

What is Tan 390 Degrees?

Tan 390 degrees is the value of tangent trigonometric function for an angle equal to 390 degrees. The value of tan 390° is 1/√3 or 0.5774 (approx).

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

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

sin(390°)/cos(390°)
± sin 390°/√(1 - sin²(390°))
± √(1 - cos²(390°))/cos 390°
± 1/√(cosec²(390°) - 1)
± √(sec²(390°) - 1)
1/cot 390°

☛ Also check: trigonometric table

What is the Value of Tan 390 Degrees in Terms of Cos 390°?

We know, using trig identities, we can write tan 390° as √(1 - cos²(390°))/cos 390°. Here, the value of cos 390° is equal to 0.866025.

What is the Exact Value of tan 390 Degrees?

The exact value of tan 390 degrees can be given accurately up to 8 decimal places as 0.57735026 or as 1/√3.

How to Find the Value of Tan 390 Degrees?

The value of tan 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 tan 390° is equal to the y-coordinate(0.5) divided by the x-coordinate (0.866). ∴ tan 390° = 1/√3 or 0.5774

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