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Tan 2000 Degrees

The value of tan 2000 degrees is 0.3639702. . .. Tan 2000 degrees in radians is written as tan (2000° × π/180°), i.e., tan (100π/9) or tan (34.906585. . .). In this article, we will discuss the methods to find the value of tan 2000 degrees with examples.

Tan 2000° in decimal: 0.3639702. . .
Tan (-2000 degrees): -0.3639702. . .
Tan 2000° in radians: tan (100π/9) or tan (34.9065850 . . .)

What is the Value of Tan 2000 Degrees?

The value of tan 2000 degrees in decimal is 0.363970234. . .. Tan 2000 degrees can also be expressed using the equivalent of the given angle (2000 degrees) in radians (34.90658 . . .)

We know, using degree to radian conversion, θ in radians = θ in degrees × (pi/180°)
⇒ 2000 degrees = 2000° × (π/180°) rad = 100π/9 or 34.9065 . . .
∴ tan 2000° = tan(34.9065) = 0.3639702. . .

Explanation:

For tan 2000°, the angle 2000° > 360°. We can represent tan 2000° as, tan(2000° mod 360°) = tan(200°). The angle 2000°, coterminal to angle 200°, is located in the Third Quadrant(Quadrant III).
Since tangent function is positive in the 3rd quadrant, thus tan 2000 degrees value = 0.3639702. . .
Similarly, given the periodic property of tan 2000°, it can also be written as, tan 2000 degrees = (2000° + n × 180°), n ∈ Z.
⇒ tan 2000° = tan 2180° = tan 2360°, and so on.
Note: Since, tangent is an odd function, the value of tan(-2000°) = -tan(2000°).

Methods to Find Value of Tan 2000 Degrees

The tangent function is positive in the 3rd quadrant. The value of tan 2000° is given as 0.36397. . .. We can find the value of tan 2000 degrees by:

Using Trigonometric Functions
Using Unit Circle

Tan 2000° in Terms of Trigonometric Functions

Using trigonometry formulas, we can represent the tan 2000 degrees as:

sin(2000°)/cos(2000°)
± sin 2000°/√(1 - sin²(2000°))
± √(1 - cos²(2000°))/cos 2000°
± 1/√(cosec²(2000°) - 1)
± √(sec²(2000°) - 1)
1/cot 2000°

Note: Since 2000° lies in the 3rd Quadrant, the final value of tan 2000° will be positive.

We can use trigonometric identities to represent tan 2000° as,

cot(90° - 2000°) = cot(-1910°)
-cot(90° + 2000°) = -cot 2090°
-tan (180° - 2000°) = -tan(-1820°)

Tan 2000 Degrees Using Unit Circle

To find the value of tan 2000 degrees using the unit circle, represent 2000° in the form (5 × 360°) + 200° [∵ 2000°>360°] ∵ The angle 2000° is coterminal to 200° angle and also tangent is a periodic function, tan 2000° = tan 200°.

Rotate ‘r’ anticlockwise to form 200° or 2000° angle with the positive x-axis.
The tan of 2000 degrees equals the y-coordinate(-0.342) divided by x-coordinate(-0.9397) of the point of intersection (-0.9397, -0.342) of unit circle and r.

Hence the value of tan 2000° = y/x = 0.364 (approx).

☛ Also Check:

Examples Using Tan 2000 Degrees

Example 1: Using the value of tan 2000°, solve: (sec²(2000°) - 1).

Solution:
We know, (sec²(2000°) - 1) = (tan²(2000°)) = 0.1325
⇒ (sec²(2000°) - 1) = 0.1325
Example 2: Find the value of (2 sin (1000°) cos (1000°) sec (2000°)). [Hint: Use tan 2000° = 0.364]

Solution:

Using sin 2a formula,
2 sin (1000°) cos (1000°) = sin (2 × 1000°) = sin 2000°
⇒ 2 sin (1000°) cos (1000°) sec(2000°) = sin 2000° sec 2000°
= sin 2000°/cos 2000° = tan 2000°
⇒ (2 sin (1000°) cos (1000°) sec(2000°)) = 0.364
Example 3: Simplify: 5 (tan 2000°/cot(-1910°))

Solution:

We know tan 2000° = cot(-1910°)
⇒ 5 tan 2000°/cot(-1910°) = 5 (tan 2000°/tan 2000°)
= 5(1) = 5

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FAQs on Tan 2000 Degrees

What is Tan 2000 Degrees?

Tan 2000 degrees is the value of tangent trigonometric function for an angle equal to 2000 degrees. The value of tan 2000° is 0.364 (approx).

How to Find Tan 2000° in Terms of Other Trigonometric Functions?

Using trigonometry formula, the value of tan 2000° can be given in terms of other trigonometric functions as:

sin(2000°)/cos(2000°)
± sin 2000°/√(1 - sin²(2000°))
± √(1 - cos²(2000°))/cos 2000°
± 1/√(cosec²(2000°) - 1)
± √(sec²(2000°) - 1)
1/cot 2000°

☛ Also check: trigonometric table

How to Find the Value of Tan 2000 Degrees?

The value of tan 2000 degrees can be calculated by constructing an angle of 2000° with the x-axis, and then finding the coordinates of the corresponding point (-0.9397, -0.342) on the unit circle. The value of tan 2000° is equal to the y-coordinate(-0.342) divided by the x-coordinate (-0.9397). ∴ tan 2000° = 0.364

What is the Value of Tan 2000 Degrees in Terms of Cot 2000°?

Since the tangent function is the reciprocal of the cotangent function, we can write tan 2000° as 1/cot(2000°). The value of cot 2000° is equal to 2.74747.

What is the Value of Tan 2000° in Terms of Sec 2000°?

We can represent the tangent function in terms of the secant function using trig identities, tan 2000° can be written as √(sec²(2000°) - 1). Here, the value of sec 2000° is equal to -1.0641.

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