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Tan 9 Degrees

The value of tan 9 degrees is 0.1583844. . .. Tan 9 degrees in radians is written as tan (9° × π/180°), i.e., tan (π/20) or tan (0.157079. . .). In this article, we will discuss the methods to find the value of tan 9 degrees with examples.

Tan 9° in decimal: 0.1583844. . .
Tan (-9 degrees): -0.1583844. . .
Tan 9° in radians: tan (π/20) or tan (0.1570796 . . .)

What is the Value of Tan 9 Degrees?

The value of tan 9 degrees in decimal is 0.158384440. . .. Tan 9 degrees can also be expressed using the equivalent of the given angle (9 degrees) in radians (0.15707 . . .)

We know, using degree to radian conversion, θ in radians = θ in degrees × (pi/180°)
⇒ 9 degrees = 9° × (π/180°) rad = π/20 or 0.1570 . . .
∴ tan 9° = tan(0.1570) = 0.1583844. . .

Explanation:

For tan 9 degrees, the angle 9° lies between 0° and 90° (First Quadrant). Since tangent function is positive in the first quadrant, thus tan 9° value = 0.1583844. . .
Since the tangent function is a periodic function, we can represent tan 9° as, tan 9 degrees = tan(9° + n × 180°), n ∈ Z.
⇒ tan 9° = tan 189° = tan 369°, and so on.
Note: Since, tangent is an odd function, the value of tan(-9°) = -tan(9°).

Methods to Find Value of Tan 9 Degrees

The tangent function is positive in the 1st quadrant. The value of tan 9° is given as 0.15838. . .. We can find the value of tan 9 degrees by:

Using Trigonometric Functions
Using Unit Circle

Tan 9° in Terms of Trigonometric Functions

Using trigonometry formulas, we can represent the tan 9 degrees as:

sin(9°)/cos(9°)
± sin 9°/√(1 - sin²(9°))
± √(1 - cos²(9°))/cos 9°
± 1/√(cosec²(9°) - 1)
± √(sec²(9°) - 1)
1/cot 9°

Note: Since 9° lies in the 1st Quadrant, the final value of tan 9° will be positive.

We can use trigonometric identities to represent tan 9° as,

cot(90° - 9°) = cot 81°
-cot(90° + 9°) = -cot 99°
-tan (180° - 9°) = -tan 171°

Tan 9 Degrees Using Unit Circle

To find the value of tan 9 degrees using the unit circle:

Rotate ‘r’ anticlockwise to form 9° angle with the positive x-axis.
The tan of 9 degrees equals the y-coordinate(0.1564) divided by x-coordinate(0.9877) of the point of intersection (0.9877, 0.1564) of unit circle and r.

Hence the value of tan 9° = y/x = 0.1584 (approx).

☛ Also Check:

Examples Using Tan 9 Degrees

Example 1: Find the value of tan 9° if cot 9° is 6.3137.

Solution:
Since, tan 9° = 1/cot 9°
⇒ tan 9° = 1/6.3137 = 0.1584
Example 2: Find the value of (2 sin (4.5°) cos (4.5°) sec (9°)). [Hint: Use tan 9° = 0.1584]

Solution:

Using sin 2a formula,
2 sin (4.5°) cos (4.5°) = sin (2 × 4.5°) = sin 9°
⇒ 2 sin (4.5°) cos (4.5°) sec(9°) = sin 9° sec 9°
= sin 9°/cos 9° = tan 9°
⇒ (2 sin (4.5°) cos (4.5°) sec(9°)) = 0.1584
Example 3: Find the value of 2 tan(9°)/5 tan(171°).

Solution:

Using trigonometric identities, we know, tan(9°) = -tan(180° - 9°) = -tan 171°.
⇒ tan(9°) = -tan(171°)
⇒ Value of 2 tan(9°)/5 tan(171°) = -2/5

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FAQs on Tan 9 Degrees

What is Tan 9 Degrees?

Tan 9 degrees is the value of tangent trigonometric function for an angle equal to 9 degrees. The value of tan 9° is 0.1584 (approx).

What is the Value of Tan 9° in Terms of Cosec 9°?

Since the tangent function can be represented using the cosecant function, we can write tan 9° as 1/√(cosec²(9°) - 1). The value of cosec 9° is equal to 6.39245.

What is the Value of Tan 9 Degrees in Terms of Cot 9°?

Since the tangent function is the reciprocal of the cotangent function, we can write tan 9° as 1/cot(9°). The value of cot 9° is equal to 6.31375.

How to Find Tan 9° in Terms of Other Trigonometric Functions?

Using trigonometry formula, the value of tan 9° can be given in terms of other trigonometric functions as:

sin(9°)/cos(9°)
± sin 9°/√(1 - sin²(9°))
± √(1 - cos²(9°))/cos 9°
± 1/√(cosec²(9°) - 1)
± √(sec²(9°) - 1)
1/cot 9°

☛ Also check: trigonometry table

How to Find the Value of Tan 9 Degrees?

The value of tan 9 degrees can be calculated by constructing an angle of 9° with the x-axis, and then finding the coordinates of the corresponding point (0.9877, 0.1564) on the unit circle. The value of tan 9° is equal to the y-coordinate(0.1564) divided by the x-coordinate (0.9877). ∴ tan 9° = 0.1584

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