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Tan 4 Degrees

The value of tan 4 degrees is 0.0699268. . .. Tan 4 degrees in radians is written as tan (4° × π/180°), i.e., tan (π/45) or tan (0.069813. . .). In this article, we will discuss the methods to find the value of tan 4 degrees with examples.

Tan 4° in decimal: 0.0699268. . .
Tan (-4 degrees): -0.0699268. . .
Tan 4° in radians: tan (π/45) or tan (0.0698131 . . .)

What is the Value of Tan 4 Degrees?

The value of tan 4 degrees in decimal is 0.069926811. . .. Tan 4 degrees can also be expressed using the equivalent of the given angle (4 degrees) in radians (0.06981 . . .)

We know, using degree to radian conversion, θ in radians = θ in degrees × (pi/180°)
⇒ 4 degrees = 4° × (π/180°) rad = π/45 or 0.0698 . . .
∴ tan 4° = tan(0.0698) = 0.0699268. . .

Explanation:

For tan 4 degrees, the angle 4° lies between 0° and 90° (First Quadrant). Since tangent function is positive in the first quadrant, thus tan 4° value = 0.0699268. . .
Since the tangent function is a periodic function, we can represent tan 4° as, tan 4 degrees = tan(4° + n × 180°), n ∈ Z.
⇒ tan 4° = tan 184° = tan 364°, and so on.
Note: Since, tangent is an odd function, the value of tan(-4°) = -tan(4°).

Methods to Find Value of Tan 4 Degrees

The tangent function is positive in the 1st quadrant. The value of tan 4° is given as 0.06992. . .. We can find the value of tan 4 degrees by:

Using Unit Circle
Using Trigonometric Functions

Tan 4 Degrees Using Unit Circle

To find the value of tan 4 degrees using the unit circle:

Rotate ‘r’ anticlockwise to form 4° angle with the positive x-axis.
The tan of 4 degrees equals the y-coordinate(0.0698) divided by x-coordinate(0.9976) of the point of intersection (0.9976, 0.0698) of unit circle and r.

Hence the value of tan 4° = y/x = 0.0699 (approx).

Tan 4° in Terms of Trigonometric Functions

Using trigonometry formulas, we can represent the tan 4 degrees as:

sin(4°)/cos(4°)
± sin 4°/√(1 - sin²(4°))
± √(1 - cos²(4°))/cos 4°
± 1/√(cosec²(4°) - 1)
± √(sec²(4°) - 1)
1/cot 4°

Note: Since 4° lies in the 1st Quadrant, the final value of tan 4° will be positive.

We can use trigonometric identities to represent tan 4° as,

cot(90° - 4°) = cot 86°
-cot(90° + 4°) = -cot 94°
-tan (180° - 4°) = -tan 176°

☛ Also Check:

Examples Using Tan 4 Degrees

Example 1: Using the value of tan 4°, solve: (sec²(4°) - 1).

Solution:
We know, (sec²(4°) - 1) = (tan²(4°)) = 0.0049
⇒ (sec²(4°) - 1) = 0.0049
Example 2: Find the value of (2 sin (2°) cos (2°) sec (4°)). [Hint: Use tan 4° = 0.0699]

Solution:

Using sin 2a formula,
2 sin (2°) cos (2°) = sin (2 × 2°) = sin 4°
⇒ 2 sin (2°) cos (2°) sec(4°) = sin 4° sec 4°
= sin 4°/cos 4° = tan 4°
⇒ (2 sin (2°) cos (2°) sec(4°)) = 0.0699
Example 3: Simplify: 8 (tan 4°/cot 86°)

Solution:

We know tan 4° = cot 86°
⇒ 8 tan 4°/cot 86° = 8 (tan 4°/tan 4°)
= 8(1) = 8

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FAQs on Tan 4 Degrees

What is Tan 4 Degrees?

Tan 4 degrees is the value of tangent trigonometric function for an angle equal to 4 degrees. The value of tan 4° is 0.0699 (approx).

How to Find the Value of Tan 4 Degrees?

The value of tan 4 degrees can be calculated by constructing an angle of 4° with the x-axis, and then finding the coordinates of the corresponding point (0.9976, 0.0698) on the unit circle. The value of tan 4° is equal to the y-coordinate(0.0698) divided by the x-coordinate (0.9976). ∴ tan 4° = 0.0699

How to Find Tan 4° in Terms of Other Trigonometric Functions?

Using trigonometry formula, the value of tan 4° can be given in terms of other trigonometric functions as:

sin(4°)/cos(4°)
± sin 4°/√(1 - sin²(4°))
± √(1 - cos²(4°))/cos 4°
± 1/√(cosec²(4°) - 1)
± √(sec²(4°) - 1)
1/cot 4°

☛ Also check: trigonometric table

What is the Value of Tan 4° in Terms of Cosec 4°?

Since the tangent function can be represented using the cosecant function, we can write tan 4° as 1/√(cosec²(4°) - 1). The value of cosec 4° is equal to 14.33558.

What is the Value of Tan 4 Degrees in Terms of Cot 4°?

Since the tangent function is the reciprocal of the cotangent function, we can write tan 4° as 1/cot(4°). The value of cot 4° is equal to 14.30066.

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