Tan 165 Degrees

The value of tan 165 degrees is -0.2679491. . .. Tan 165 degrees in radians is written as tan (165° × π/180°), i.e., tan (11π/12) or tan (2.879793. . .). In this article, we will discuss the methods to find the value of tan 165 degrees with examples.

• Tan 165°: -2 + √3
• Tan 165° in decimal: -0.2679491. . .
• Tan (-165 degrees): 0.2679491. . . or 2 - √3
• Tan 165° in radians: tan (11π/12) or tan (2.8797932 . . .)

What is the Value of Tan 165 Degrees?

The value of tan 165 degrees in decimal is -0.267949192. . .. Tan 165 degrees can also be expressed using the equivalent of the given angle (165 degrees) in radians (2.87979 . . .)

We know, using degree to radian conversion, θ in radians = θ in degrees × (pi/180°)
⇒ 165 degrees = 165° × (π/180°) rad = 11π/12 or 2.8797 . . .
∴ tan 165° = tan(2.8797) = -2 + √3 or -0.2679491. . .

Explanation:

For tan 165 degrees, the angle 165° lies between 90° and 180° (Second Quadrant). Since tangent function is negative in the second quadrant, thus tan 165° value = -2 + √3 or -0.2679491. . .
Since the tangent function is a periodic function, we can represent tan 165° as, tan 165 degrees = tan(165° + n × 180°), n ∈ Z.
⇒ tan 165° = tan 345° = tan 525°, and so on.
Note: Since, tangent is an odd function, the value of tan(-165°) = -tan(165°).

Methods to Find Value of Tan 165 Degrees

The tangent function is negative in the 2nd quadrant. The value of tan 165° is given as -0.26794. . .. We can find the value of tan 165 degrees by:

• Using Unit Circle
• Using Trigonometric Functions

Tan 165 Degrees Using Unit Circle

To find the value of tan 165 degrees using the unit circle:

• Rotate ‘r’ anticlockwise to form 165° angle with the positive x-axis.
• The tan of 165 degrees equals the y-coordinate(0.2588) divided by x-coordinate(-0.9659) of the point of intersection (-0.9659, 0.2588) of unit circle and r.

Hence the value of tan 165° = y/x = -0.2679 (approx).

Tan 165° in Terms of Trigonometric Functions

Using trigonometry formulas, we can represent the tan 165 degrees as:

• sin(165°)/cos(165°)
• ± sin 165°/√(1 - sin²(165°))
• ± √(1 - cos²(165°))/cos 165°
• ± 1/√(cosec²(165°) - 1)
• ± √(sec²(165°) - 1)
• 1/cot 165°

Note: Since 165° lies in the 2nd Quadrant, the final value of tan 165° will be negative.

We can use trigonometric identities to represent tan 165° as,

• cot(90° - 165°) = cot(-75°)
• -cot(90° + 165°) = -cot 255°
• -tan (180° - 165°) = -tan 15°

☛ Also Check:

• tan 37 degrees
• tan 90 degrees
• tan 53 degrees
• tan 18 degrees
• tan 67 degrees
• tan 285 degrees

FAQs on Tan 165 Degrees

What is Tan 165 Degrees?

Tan 165 degrees is the value of tangent trigonometric function for an angle equal to 165 degrees. The value of tan 165° is -2 + √3 or -0.2679 (approx).

What is the Exact Value of tan 165 Degrees?

The exact value of tan 165 degrees can be given accurately up to 8 decimal places as -0.26794919 or as -2 + √3.

How to Find Tan 165° in Terms of Other Trigonometric Functions?

Using trigonometry formula, the value of tan 165° can be given in terms of other trigonometric functions as:

• sin(165°)/cos(165°)
• ± sin 165°/√(1 - sin²(165°))
• ± √(1 - cos²(165°))/cos 165°
• ± 1/√(cosec²(165°) - 1)
• ± √(sec²(165°) - 1)
• 1/cot 165°

☛ Also check: trigonometric table

What is the Value of Tan 165 Degrees in Terms of Cot 165°?

Since the tangent function is the reciprocal of the cotangent function, we can write tan 165° as 1/cot(165°). The value of cot 165° is equal to -(2 + √3).

How to Find the Value of Tan 165 Degrees?

The value of tan 165 degrees can be calculated by constructing an angle of 165° with the x-axis, and then finding the coordinates of the corresponding point (-0.9659, 0.2588) on the unit circle. The value of tan 165° is equal to the y-coordinate(0.2588) divided by the x-coordinate (-0.9659). ∴ tan 165° = -2 + √3 or -0.2679