Tan 585 Degrees

The value of tan 585 degrees is 1. Tan 585 degrees in radians is written as tan (585° × π/180°), i.e., tan (13π/4) or tan (10.210176. . .). In this article, we will discuss the methods to find the value of tan 585 degrees with examples.

• Tan 585°: 1
• Tan (-585 degrees): -1
• Tan 585° in radians: tan (13π/4) or tan (10.2101761 . . .)

What is the Value of Tan 585 Degrees?

The value of tan 585 degrees is 1. Tan 585 degrees can also be expressed using the equivalent of the given angle (585 degrees) in radians (10.21017 . . .)

We know, using degree to radian conversion, θ in radians = θ in degrees × (pi/180°)
⇒ 585 degrees = 585° × (π/180°) rad = 13π/4 or 10.2101 . . .
∴ tan 585° = tan(10.2101) = 1

Explanation:

For tan 585°, the angle 585° > 360°. We can represent tan 585° as, tan(585° mod 360°) = tan(225°). The angle 585°, coterminal to angle 225°, is located in the Third Quadrant(Quadrant III).
Since tangent function is positive in the 3rd quadrant, thus tan 585 degrees value = 1
Similarly, given the periodic property of tan 585°, it can also be written as, tan 585 degrees = (585° + n × 180°), n ∈ Z.
⇒ tan 585° = tan 765° = tan 945°, and so on.
Note: Since, tangent is an odd function, the value of tan(-585°) = -tan(585°).

Methods to Find Value of Tan 585 Degrees

The tangent function is positive in the 3rd quadrant. The value of tan 585° is given as 1. We can find the value of tan 585 degrees by:

• Using Unit Circle
• Using Trigonometric Functions

Tan 585 Degrees Using Unit Circle

To find the value of tan 585 degrees using the unit circle, represent 585° in the form (1 × 360°) + 225° [∵ 585°>360°] ∵ The angle 585° is coterminal to 225° angle and also tangent is a periodic function, tan 585° = tan 225°.

• Rotate ‘r’ anticlockwise to form 225° or 585° angle with the positive x-axis.
• The tan of 585 degrees equals the y-coordinate(-0.7071) divided by x-coordinate(-0.7071) of the point of intersection (-0.7071, -0.7071) of unit circle and r.

Hence the value of tan 585° = y/x = 1

Tan 585° in Terms of Trigonometric Functions

Using trigonometry formulas, we can represent the tan 585 degrees as:

• sin(585°)/cos(585°)
• ± sin 585°/√(1 - sin²(585°))
• ± √(1 - cos²(585°))/cos 585°
• ± 1/√(cosec²(585°) - 1)
• ± √(sec²(585°) - 1)
• 1/cot 585°

Note: Since 585° lies in the 3rd Quadrant, the final value of tan 585° will be positive.

We can use trigonometric identities to represent tan 585° as,

• cot(90° - 585°) = cot(-495°)
• -cot(90° + 585°) = -cot 675°
• -tan (180° - 585°) = -tan(-405°)

☛ Also Check:

• tan 690 degrees
• tan 12 degrees
• tan 2 degrees
• tan 8 degrees
• tan 31 degrees
• tan 365 degrees

FAQs on Tan 585 Degrees

What is Tan 585 Degrees?

Tan 585 degrees is the value of tangent trigonometric function for an angle equal to 585 degrees. The value of tan 585° is 1.

What is the Value of Tan 585° in Terms of Cosec 585°?

Since the tangent function can be represented using the cosecant function, we can write tan 585° as 1/√(cosec²(585°) - 1). The value of cosec 585° is equal to -1.41421.

How to Find the Value of Tan 585 Degrees?

The value of tan 585 degrees can be calculated by constructing an angle of 585° with the x-axis, and then finding the coordinates of the corresponding point (-0.7071, -0.7071) on the unit circle. The value of tan 585° is equal to the y-coordinate(-0.7071) divided by the x-coordinate (-0.7071). ∴ tan 585° = 1

How to Find Tan 585° in Terms of Other Trigonometric Functions?

Using trigonometry formula, the value of tan 585° can be given in terms of other trigonometric functions as:

• sin(585°)/cos(585°)
• ± sin 585°/√(1 - sin²(585°))
• ± √(1 - cos²(585°))/cos 585°
• ± 1/√(cosec²(585°) - 1)
• ± √(sec²(585°) - 1)
• 1/cot 585°

☛ Also check: trigonometric table

What is the Value of Tan 585 Degrees in Terms of Sin 585°?

Using trigonometric identities, we can write tan 585° in terms of sin 585° as, tan(585°) = -sin 585°/√(1 - sin²(585°)) . Here, the value of sin 585° is equal to -0.7071.