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Tan 420 Degrees

The value of tan 420 degrees is 1.7320508. . .. Tan 420 degrees in radians is written as tan (420° × π/180°), i.e., tan (7π/3) or tan (7.330382. . .). In this article, we will discuss the methods to find the value of tan 420 degrees with examples.

Tan 420°: √3
Tan 420° in decimal: 1.7320508. . .
Tan (-420 degrees): -1.7320508. . . or -√3
Tan 420° in radians: tan (7π/3) or tan (7.3303828 . . .)

What is the Value of Tan 420 Degrees?

The value of tan 420 degrees in decimal is 1.732050807. . .. Tan 420 degrees can also be expressed using the equivalent of the given angle (420 degrees) in radians (7.33038 . . .)

We know, using degree to radian conversion, θ in radians = θ in degrees × (pi/180°)
⇒ 420 degrees = 420° × (π/180°) rad = 7π/3 or 7.3303 . . .
∴ tan 420° = tan(7.3303) = √3 or 1.7320508. . .

Explanation:

For tan 420°, the angle 420° > 360°. We can represent tan 420° as, tan(420° mod 360°) = tan(60°). The angle 420°, coterminal to angle 60°, is located in the First Quadrant(Quadrant I).
Since tangent function is positive in the 1st quadrant, thus tan 420 degrees value = √3 or 1.7320508. . .
Similarly, given the periodic property of tan 420°, it can also be written as, tan 420 degrees = (420° + n × 180°), n ∈ Z.
⇒ tan 420° = tan 600° = tan 780°, and so on.
Note: Since, tangent is an odd function, the value of tan(-420°) = -tan(420°).

Methods to Find Value of Tan 420 Degrees

The tangent function is positive in the 1st quadrant. The value of tan 420° is given as 1.73205. . .. We can find the value of tan 420 degrees by:

Using Trigonometric Functions
Using Unit Circle

Tan 420° in Terms of Trigonometric Functions

Using trigonometry formulas, we can represent the tan 420 degrees as:

sin(420°)/cos(420°)
± sin 420°/√(1 - sin²(420°))
± √(1 - cos²(420°))/cos 420°
± 1/√(cosec²(420°) - 1)
± √(sec²(420°) - 1)
1/cot 420°

Note: Since 420° lies in the 1st Quadrant, the final value of tan 420° will be positive.

We can use trigonometric identities to represent tan 420° as,

cot(90° - 420°) = cot(-330°)
-cot(90° + 420°) = -cot 510°
-tan (180° - 420°) = -tan(-240°)

Tan 420 Degrees Using Unit Circle

To find the value of tan 420 degrees using the unit circle, represent 420° in the form (1 × 360°) + 60° [∵ 420°>360°] ∵ The angle 420° is coterminal to 60° angle and also tangent is a periodic function, tan 420° = tan 60°.

Rotate ‘r’ anticlockwise to form 60° or 420° angle with the positive x-axis.
The tan of 420 degrees equals the y-coordinate(0.866) divided by x-coordinate(0.5) of the point of intersection (0.5, 0.866) of unit circle and r.

Hence the value of tan 420° = y/x = 1.7321 (approx).

☛ Also Check:

Examples Using Tan 420 Degrees

Example 1: Using the value of tan 420°, solve: (sec²(420°) - 1).

Solution:
We know, (sec²(420°) - 1) = (tan²(420°)) = 3
⇒ (sec²(420°) - 1) = 3
Example 2: Find the value of 3 tan(420°)/10 tan(-240°).

Solution:

Using trigonometric identities, we know, tan(420°) = -tan(180° - 420°) = -tan(-240°).
⇒ tan(420°) = -tan(-240°)
⇒ Value of 3 tan(420°)/10 tan(-240°) = -3/10
Example 3: Find the value of 2 tan 210°/(1 - tan²(210°)). [Hint: Use tan 420° = 1.7321]

Solution:

Using the tan 2a formula,
2 tan 210°/(1 - tan²(210°)) = tan(2 × 210°) = tan 420°
∵ tan 420° = 1.7321
⇒ 2 tan 210°/(1 - tan²(210°)) = 1.7321

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FAQs on Tan 420 Degrees

What is Tan 420 Degrees?

Tan 420 degrees is the value of tangent trigonometric function for an angle equal to 420 degrees. The value of tan 420° is √3 or 1.7321 (approx).

How to Find the Value of Tan 420 Degrees?

The value of tan 420 degrees can be calculated by constructing an angle of 420° with the x-axis, and then finding the coordinates of the corresponding point (0.5, 0.866) on the unit circle. The value of tan 420° is equal to the y-coordinate(0.866) divided by the x-coordinate (0.5). ∴ tan 420° = √3 or 1.7321

What is the Value of Tan 420° in Terms of Sec 420°?

We can represent the tangent function in terms of the secant function using trig identities, tan 420° can be written as √(sec²(420°) - 1). Here, the value of sec 420° is equal to 2.

What is the Value of Tan 420 Degrees in Terms of Cos 420°?

We know, using trig identities, we can write tan 420° as √(1 - cos²(420°))/cos 420°. Here, the value of cos 420° is equal to 0.5.

How to Find Tan 420° in Terms of Other Trigonometric Functions?

Using trigonometry formula, the value of tan 420° can be given in terms of other trigonometric functions as:

sin(420°)/cos(420°)
± sin 420°/√(1 - sin²(420°))
± √(1 - cos²(420°))/cos 420°
± 1/√(cosec²(420°) - 1)
± √(sec²(420°) - 1)
1/cot 420°

☛ Also check: trigonometric table

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