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Tan 10 Degrees

The value of tan 10 degrees is 0.1763269. . .. Tan 10 degrees in radians is written as tan (10° × π/180°), i.e., tan (π/18) or tan (0.174532. . .). In this article, we will discuss the methods to find the value of tan 10 degrees with examples.

Tan 10° in decimal: 0.1763269. . .
Tan (-10 degrees): -0.1763269. . .
Tan 10° in radians: tan (π/18) or tan (0.1745329 . . .)

What is the Value of Tan 10 Degrees?

The value of tan 10 degrees in decimal is 0.176326980. . .. Tan 10 degrees can also be expressed using the equivalent of the given angle (10 degrees) in radians (0.17453 . . .)

We know, using degree to radian conversion, θ in radians = θ in degrees × (pi/180°)
⇒ 10 degrees = 10° × (π/180°) rad = π/18 or 0.1745 . . .
∴ tan 10° = tan(0.1745) = 0.1763269. . .

Explanation:

For tan 10 degrees, the angle 10° lies between 0° and 90° (First Quadrant). Since tangent function is positive in the first quadrant, thus tan 10° value = 0.1763269. . .
Since the tangent function is a periodic function, we can represent tan 10° as, tan 10 degrees = tan(10° + n × 180°), n ∈ Z.
⇒ tan 10° = tan 190° = tan 370°, and so on.
Note: Since, tangent is an odd function, the value of tan(-10°) = -tan(10°).

Methods to Find Value of Tan 10 Degrees

The tangent function is positive in the 1st quadrant. The value of tan 10° is given as 0.17632. . .. We can find the value of tan 10 degrees by:

Using Unit Circle
Using Trigonometric Functions

Tan 10 Degrees Using Unit Circle

To find the value of tan 10 degrees using the unit circle:

Rotate ‘r’ anticlockwise to form 10° angle with the positive x-axis.
The tan of 10 degrees equals the y-coordinate(0.1736) divided by x-coordinate(0.9848) of the point of intersection (0.9848, 0.1736) of unit circle and r.

Hence the value of tan 10° = y/x = 0.1763 (approx).

Tan 10° in Terms of Trigonometric Functions

Using trigonometry formulas, we can represent the tan 10 degrees as:

sin(10°)/cos(10°)
± sin 10°/√(1 - sin²(10°))
± √(1 - cos²(10°))/cos 10°
± 1/√(cosec²(10°) - 1)
± √(sec²(10°) - 1)
1/cot 10°

Note: Since 10° lies in the 1st Quadrant, the final value of tan 10° will be positive.

We can use trigonometric identities to represent tan 10° as,

cot(90° - 10°) = cot 80°
-cot(90° + 10°) = -cot 100°
-tan (180° - 10°) = -tan 170°

☛ Also Check:

Examples Using Tan 10 Degrees

Example 1: Using the value of tan 10°, solve: (sec²(10°) - 1).

Solution:
We know, (sec²(10°) - 1) = (tan²(10°)) = 0.0311
⇒ (sec²(10°) - 1) = 0.0311
Example 2: Find the value of 6 tan(10°)/9 tan(170°).

Solution:

Using trigonometric identities, we know, tan(10°) = -tan(180° - 10°) = -tan 170°.
⇒ tan(10°) = -tan(170°)
⇒ Value of 6 tan(10°)/9 tan(170°) = -6/9 = -2/3
Example 3: Find the value of (2 sin (5°) cos (5°) sec (10°)). [Hint: Use tan 10° = 0.1763]

Solution:

Using sin 2a formula,
2 sin (5°) cos (5°) = sin (2 × 5°) = sin 10°
⇒ 2 sin (5°) cos (5°) sec(10°) = sin 10° sec 10°
= sin 10°/cos 10° = tan 10°
⇒ (2 sin (5°) cos (5°) sec(10°)) = 0.1763

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FAQs on Tan 10 Degrees

What is Tan 10 Degrees?

Tan 10 degrees is the value of tangent trigonometric function for an angle equal to 10 degrees. The value of tan 10° is 0.1763 (approx).

What is the Value of Tan 10 Degrees in Terms of Cos 10°?

We know, using trig identities, we can write tan 10° as √(1 - cos²(10°))/cos 10°. Here, the value of cos 10° is equal to 0.984807.

What is the Value of Tan 10° in Terms of Cosec 10°?

Since the tangent function can be represented using the cosecant function, we can write tan 10° as 1/√(cosec²(10°) - 1). The value of cosec 10° is equal to 5.75877.

How to Find the Value of Tan 10 Degrees?

The value of tan 10 degrees can be calculated by constructing an angle of 10° with the x-axis, and then finding the coordinates of the corresponding point (0.9848, 0.1736) on the unit circle. The value of tan 10° is equal to the y-coordinate(0.1736) divided by the x-coordinate (0.9848). ∴ tan 10° = 0.1763

How to Find Tan 10° in Terms of Other Trigonometric Functions?

Using trigonometry formula, the value of tan 10° can be given in terms of other trigonometric functions as:

sin(10°)/cos(10°)
± sin 10°/√(1 - sin²(10°))
± √(1 - cos²(10°))/cos 10°
± 1/√(cosec²(10°) - 1)
± √(sec²(10°) - 1)
1/cot 10°

☛ Also check: trigonometric table

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