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Tan 22 Degrees

The value of tan 22 degrees is 0.4040262. . .. Tan 22 degrees in radians is written as tan (22° × π/180°), i.e., tan (11π/90) or tan (0.383972. . .). In this article, we will discuss the methods to find the value of tan 22 degrees with examples.

Tan 22° in decimal: 0.4040262. . .
Tan (-22 degrees): -0.4040262. . .
Tan 22° in radians: tan (11π/90) or tan (0.3839724 . . .)

What is the Value of Tan 22 Degrees?

The value of tan 22 degrees in decimal is 0.404026225. . .. Tan 22 degrees can also be expressed using the equivalent of the given angle (22 degrees) in radians (0.38397 . . .)

We know, using degree to radian conversion, θ in radians = θ in degrees × (pi/180°)
⇒ 22 degrees = 22° × (π/180°) rad = 11π/90 or 0.3839 . . .
∴ tan 22° = tan(0.3839) = 0.4040262. . .

Explanation:

For tan 22 degrees, the angle 22° lies between 0° and 90° (First Quadrant). Since tangent function is positive in the first quadrant, thus tan 22° value = 0.4040262. . .
Since the tangent function is a periodic function, we can represent tan 22° as, tan 22 degrees = tan(22° + n × 180°), n ∈ Z.
⇒ tan 22° = tan 202° = tan 382°, and so on.
Note: Since, tangent is an odd function, the value of tan(-22°) = -tan(22°).

Methods to Find Value of Tan 22 Degrees

The tangent function is positive in the 1st quadrant. The value of tan 22° is given as 0.40402. . .. We can find the value of tan 22 degrees by:

Using Unit Circle
Using Trigonometric Functions

Tan 22 Degrees Using Unit Circle

To find the value of tan 22 degrees using the unit circle:

Rotate ‘r’ anticlockwise to form 22° angle with the positive x-axis.
The tan of 22 degrees equals the y-coordinate(0.3746) divided by x-coordinate(0.9272) of the point of intersection (0.9272, 0.3746) of unit circle and r.

Hence the value of tan 22° = y/x = 0.404 (approx).

Tan 22° in Terms of Trigonometric Functions

Using trigonometry formulas, we can represent the tan 22 degrees as:

sin(22°)/cos(22°)
± sin 22°/√(1 - sin²(22°))
± √(1 - cos²(22°))/cos 22°
± 1/√(cosec²(22°) - 1)
± √(sec²(22°) - 1)
1/cot 22°

Note: Since 22° lies in the 1st Quadrant, the final value of tan 22° will be positive.

We can use trigonometric identities to represent tan 22° as,

cot(90° - 22°) = cot 68°
-cot(90° + 22°) = -cot 112°
-tan (180° - 22°) = -tan 158°

☛ Also Check:

Examples Using Tan 22 Degrees

Example 1: Find the value of (2 sin (11°) cos (11°) sec (22°)). [Hint: Use tan 22° = 0.404]

Solution:

Using sin 2a formula,
2 sin (11°) cos (11°) = sin (2 × 11°) = sin 22°
⇒ 2 sin (11°) cos (11°) sec(22°) = sin 22° sec 22°
= sin 22°/cos 22° = tan 22°
⇒ (2 sin (11°) cos (11°) sec(22°)) = 0.404
Example 2: Find the value of 5 tan(22°)/9 tan(158°).

Solution:

Using trigonometric identities, we know, tan(22°) = -tan(180° - 22°) = -tan 158°.
⇒ tan(22°) = -tan(158°)
⇒ Value of 5 tan(22°)/9 tan(158°) = -5/9
Example 3: Simplify: 6 (tan 22°/cot 68°)

Solution:

We know tan 22° = cot 68°
⇒ 6 tan 22°/cot 68° = 6 (tan 22°/tan 22°)
= 6(1) = 6

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FAQs on Tan 22 Degrees

What is Tan 22 Degrees?

Tan 22 degrees is the value of tangent trigonometric function for an angle equal to 22 degrees. The value of tan 22° is 0.404 (approx).

What is the Exact Value of tan 22 Degrees?

The exact value of tan 22 degrees can be given accurately up to 8 decimal places as 0.40402622.

How to Find Tan 22° in Terms of Other Trigonometric Functions?

Using trigonometry formula, the value of tan 22° can be given in terms of other trigonometric functions as:

sin(22°)/cos(22°)
± sin 22°/√(1 - sin²(22°))
± √(1 - cos²(22°))/cos 22°
± 1/√(cosec²(22°) - 1)
± √(sec²(22°) - 1)
1/cot 22°

☛ Also check: trigonometric table

What is the Value of Tan 22 Degrees in Terms of Cos 22°?

We know, using trig identities, we can write tan 22° as √(1 - cos²(22°))/cos 22°. Here, the value of cos 22° is equal to 0.927183.

How to Find the Value of Tan 22 Degrees?

The value of tan 22 degrees can be calculated by constructing an angle of 22° with the x-axis, and then finding the coordinates of the corresponding point (0.9272, 0.3746) on the unit circle. The value of tan 22° is equal to the y-coordinate(0.3746) divided by the x-coordinate (0.9272). ∴ tan 22° = 0.404

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