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Tan 5 Degrees

The value of tan 5 degrees is 0.0874886. . .. Tan 5 degrees in radians is written as tan (5° × π/180°), i.e., tan (π/36) or tan (0.087266. . .). In this article, we will discuss the methods to find the value of tan 5 degrees with examples.

Tan 5° in decimal: 0.0874886. . .
Tan (-5 degrees): -0.0874886. . .
Tan 5° in radians: tan (π/36) or tan (0.0872664 . . .)

What is the Value of Tan 5 Degrees?

The value of tan 5 degrees in decimal is 0.087488663. . .. Tan 5 degrees can also be expressed using the equivalent of the given angle (5 degrees) in radians (0.08726 . . .)

We know, using degree to radian conversion, θ in radians = θ in degrees × (pi/180°)
⇒ 5 degrees = 5° × (π/180°) rad = π/36 or 0.0872 . . .
∴ tan 5° = tan(0.0872) = 0.0874886. . .

Explanation:

For tan 5 degrees, the angle 5° lies between 0° and 90° (First Quadrant). Since tangent function is positive in the first quadrant, thus tan 5° value = 0.0874886. . .
Since the tangent function is a periodic function, we can represent tan 5° as, tan 5 degrees = tan(5° + n × 180°), n ∈ Z.
⇒ tan 5° = tan 185° = tan 365°, and so on.
Note: Since, tangent is an odd function, the value of tan(-5°) = -tan(5°).

Methods to Find Value of Tan 5 Degrees

The tangent function is positive in the 1st quadrant. The value of tan 5° is given as 0.08748. . .. We can find the value of tan 5 degrees by:

Using Unit Circle
Using Trigonometric Functions

Tan 5 Degrees Using Unit Circle

To find the value of tan 5 degrees using the unit circle:

Rotate ‘r’ anticlockwise to form 5° angle with the positive x-axis.
The tan of 5 degrees equals the y-coordinate(0.0872) divided by x-coordinate(0.9962) of the point of intersection (0.9962, 0.0872) of unit circle and r.

Hence the value of tan 5° = y/x = 0.0875 (approx).

Tan 5° in Terms of Trigonometric Functions

Using trigonometry formulas, we can represent the tan 5 degrees as:

sin(5°)/cos(5°)
± sin 5°/√(1 - sin²(5°))
± √(1 - cos²(5°))/cos 5°
± 1/√(cosec²(5°) - 1)
± √(sec²(5°) - 1)
1/cot 5°

Note: Since 5° lies in the 1st Quadrant, the final value of tan 5° will be positive.

We can use trigonometric identities to represent tan 5° as,

cot(90° - 5°) = cot 85°
-cot(90° + 5°) = -cot 95°
-tan (180° - 5°) = -tan 175°

☛ Also Check:

Examples Using Tan 5 Degrees

Example 1: Find the value of 2 tan(5°)/5 tan(175°).

Solution:

Using trigonometric identities, we know, tan(5°) = -tan(180° - 5°) = -tan 175°.
⇒ tan(5°) = -tan(175°)
⇒ Value of 2 tan(5°)/5 tan(175°) = -2/5
Example 2: Find the value of (2 sin (2.5°) cos (2.5°) sec (5°)). [Hint: Use tan 5° = 0.0875]

Solution:

Using sin 2a formula,
2 sin (2.5°) cos (2.5°) = sin (2 × 2.5°) = sin 5°
⇒ 2 sin (2.5°) cos (2.5°) sec(5°) = sin 5° sec 5°
= sin 5°/cos 5° = tan 5°
⇒ (2 sin (2.5°) cos (2.5°) sec(5°)) = 0.0875
Example 3: Using the value of tan 5°, solve: (sec²(5°) - 1).

Solution:
We know, (sec²(5°) - 1) = (tan²(5°)) = 0.0077
⇒ (sec²(5°) - 1) = 0.0077

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FAQs on Tan 5 Degrees

What is Tan 5 Degrees?

Tan 5 degrees is the value of tangent trigonometric function for an angle equal to 5 degrees. The value of tan 5° is 0.0875 (approx).

What is the Value of Tan 5 Degrees in Terms of Cot 5°?

Since the tangent function is the reciprocal of the cotangent function, we can write tan 5° as 1/cot(5°). The value of cot 5° is equal to 11.43005.

What is the Value of Tan 5° in Terms of Cosec 5°?

Since the tangent function can be represented using the cosecant function, we can write tan 5° as 1/√(cosec²(5°) - 1). The value of cosec 5° is equal to 11.47371.

How to Find the Value of Tan 5 Degrees?

The value of tan 5 degrees can be calculated by constructing an angle of 5° with the x-axis, and then finding the coordinates of the corresponding point (0.9962, 0.0872) on the unit circle. The value of tan 5° is equal to the y-coordinate(0.0872) divided by the x-coordinate (0.9962). ∴ tan 5° = 0.0875

How to Find Tan 5° in Terms of Other Trigonometric Functions?

Using trigonometry formula, the value of tan 5° can be given in terms of other trigonometric functions as:

sin(5°)/cos(5°)
± sin 5°/√(1 - sin²(5°))
± √(1 - cos²(5°))/cos 5°
± 1/√(cosec²(5°) - 1)
± √(sec²(5°) - 1)
1/cot 5°

☛ Also check: trigonometric table

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