Tan 105 Degrees

The value of tan 105 degrees is -3.7320508. . .. Tan 105 degrees in radians is written as tan (105° × π/180°), i.e., tan (7π/12) or tan (1.832595. . .). In this article, we will discuss the methods to find the value of tan 105 degrees with examples.

• Tan 105°: -2 - √3
• Tan 105° in decimal: -3.7320508. . .
• Tan (-105 degrees): 3.7320508. . . or 2 + √3
• Tan 105° in radians: tan (7π/12) or tan (1.8325957 . . .)

What is the Value of Tan 105 Degrees?

The value of tan 105 degrees in decimal is -3.732050807. . .. Tan 105 degrees can also be expressed using the equivalent of the given angle (105 degrees) in radians (1.83259 . . .)

We know, using degree to radian conversion, θ in radians = θ in degrees × (pi/180°)
⇒ 105 degrees = 105° × (π/180°) rad = 7π/12 or 1.8325 . . .
∴ tan 105° = tan(1.8325) = -2 - √3 or -3.7320508. . .

Explanation:

For tan 105 degrees, the angle 105° lies between 90° and 180° (Second Quadrant). Since tangent function is negative in the second quadrant, thus tan 105° value = -2 - √3 or -3.7320508. . .
Since the tangent function is a periodic function, we can represent tan 105° as, tan 105 degrees = tan(105° + n × 180°), n ∈ Z.
⇒ tan 105° = tan 285° = tan 465°, and so on.
Note: Since, tangent is an odd function, the value of tan(-105°) = -tan(105°).

Methods to Find Value of Tan 105 Degrees

The tangent function is negative in the 2nd quadrant. The value of tan 105° is given as -3.73205. . .. We can find the value of tan 105 degrees by:

• Using Trigonometric Functions
• Using Unit Circle

Tan 105° in Terms of Trigonometric Functions

Using trigonometry formulas, we can represent the tan 105 degrees as:

• sin(105°)/cos(105°)
• ± sin 105°/√(1 - sin²(105°))
• ± √(1 - cos²(105°))/cos 105°
• ± 1/√(cosec²(105°) - 1)
• ± √(sec²(105°) - 1)
• 1/cot 105°

Note: Since 105° lies in the 2nd Quadrant, the final value of tan 105° will be negative.

We can use trigonometric identities to represent tan 105° as,

• cot(90° - 105°) = cot(-15°)
• -cot(90° + 105°) = -cot 195°
• -tan (180° - 105°) = -tan 75°

Tan 105 Degrees Using Unit Circle

To find the value of tan 105 degrees using the unit circle:

• Rotate ‘r’ anticlockwise to form 105° angle with the positive x-axis.
• The tan of 105 degrees equals the y-coordinate(0.9659) divided by x-coordinate(-0.2588) of the point of intersection (-0.2588, 0.9659) of unit circle and r.

Hence the value of tan 105° = y/x = -3.7321 (approx).

☛ Also Check:

• tan 690 degrees
• tan 53 degrees
• tan 270 degrees
• tan 8 degrees
• tan 1050 degrees
• tan 73 degrees

FAQs on Tan 105 Degrees

What is Tan 105 Degrees?

Tan 105 degrees is the value of tangent trigonometric function for an angle equal to 105 degrees. The value of tan 105° is -2 - √3 or -3.7321 (approx).

What is the Value of Tan 105 Degrees in Terms of Cos 105°?

We know, using trig identities, we can write tan 105° as √(1 - cos²(105°))/cos 105°. Here, the value of cos 105° is equal to -0.258819.

How to Find the Value of Tan 105 Degrees?

The value of tan 105 degrees can be calculated by constructing an angle of 105° with the x-axis, and then finding the coordinates of the corresponding point (-0.2588, 0.9659) on the unit circle. The value of tan 105° is equal to the y-coordinate(0.9659) divided by the x-coordinate (-0.2588). ∴ tan 105° = -2 - √3 or -3.7321

How to Find Tan 105° in Terms of Other Trigonometric Functions?

Using trigonometry formula, the value of tan 105° can be given in terms of other trigonometric functions as:

• sin(105°)/cos(105°)
• ± sin 105°/√(1 - sin²(105°))
• ± √(1 - cos²(105°))/cos 105°
• ± 1/√(cosec²(105°) - 1)
• ± √(sec²(105°) - 1)
• 1/cot 105°

☛ Also check: trigonometric table

What is the Value of Tan 105° in Terms of Cosec 105°?

Since the tangent function can be represented using the cosecant function, we can write tan 105° as -1/√(cosec²(105°) - 1). The value of cosec 105° is equal to 1.03527.