Tan 90 Degrees

The value of tan 90 degrees is undefined (or, ∞). Tan 90 degrees in radians is written as tan (90° × π/180°), i.e., tan (π/2) or tan (1.570796. . .). In this article, we will discuss the methods to find the value of tan 90 degrees with examples.

• Tan 90°: undefined(∞)
• Tan (-90 degrees): undefined
• Tan 90° in radians: tan (π/2) or tan (1.5707963 . . .)

What is the Value of Tan 90 Degrees?

The value of tan 90 degrees is undefined(∞). Tan 90 degrees can also be expressed using the equivalent of the given angle (90 degrees) in radians (1.57079 . . .)

We know, using degree to radian conversion, θ in radians = θ in degrees × (pi/180°)
⇒ 90 degrees = 90° × (π/180°) rad = π/2 or 1.5707 . . .
∴ tan 90° = tan(1.5707) = undefined(∞)

Explanation:

For tan 90 degrees, the angle 90° lies on the positive y-axis. Thus, tan 90° value = undefined(∞)
Since the tangent function is a periodic function, we can represent tan 90° as, tan 90 degrees = tan(90° + n × 180°), n ∈ Z.
⇒ tan 90° = tan 270° = tan 450°, and so on.
Note: Since, tangent is an odd function, the value of tan(-90°) = -tan(90°) = undefined.

Methods to Find Value of Tan 90 Degrees

The value of tan 90° is given as undefined(∞). We can find the value of tan 90 degrees by:

• Using Unit Circle
• Using Trigonometric Functions

Tan 90 Degrees Using Unit Circle

To find the value of tan 90 degrees using the unit circle:

• Rotate ‘r’ anticlockwise to form 90° angle with the positive x-axis.
• The tan of 90 degrees equals the y-coordinate(1) divided by x-coordinate(0) of the point of intersection (0, 1) of unit circle and r.

Hence the value of tan 90° = y/x = undefined(∞)

Tan 90° in Terms of Trigonometric Functions

Using trigonometry formulas, we can represent the tan 90 degrees as:

• sin(90°)/cos(90°)
• ± sin 90°/√(1 - sin²(90°))
• ± √(1 - cos²(90°))/cos 90°
• ± 1/√(cosec²(90°) - 1)
• ± √(sec²(90°) - 1)
• 1/cot 90°

We can use trigonometric identities to represent tan 90° as,

• cot(90° - 90°) = cot 0°
• -cot(90° + 90°) = -cot 180°
• -tan (180° - 90°) = -tan 90°

Note: Since 90° lies on the positive y-axis, the final value of tan 90° will be undefined(∞).

☛ Also Check:

• tan 105 degrees
• tan 3 degrees
• tan 120 degrees
• tan 720 degrees
• tan 1050 degrees
• tan 5 degrees

FAQs on Tan 90 Degrees

What is Tan 90 Degrees?

Tan 90 degrees is the value of tangent trigonometric function for an angle equal to 90 degrees. The value of tan 90° is undefined(∞).

How to Find the Value of Tan 90 Degrees?

The value of tan 90 degrees can be calculated by constructing an angle of 90° with the x-axis, and then finding the coordinates of the corresponding point (0, 1) on the unit circle. The value of tan 90° is equal to the y-coordinate(1) divided by the x-coordinate (0). ∴ tan 90° = undefined(∞)

How to Find Tan 90° in Terms of Other Trigonometric Functions?

Using trigonometry formula, the value of tan 90° can be given in terms of other trigonometric functions as:

• sin(90°)/cos(90°)
• ± sin 90°/√(1 - sin²(90°))
• ± √(1 - cos²(90°))/cos 90°
• ± 1/√(cosec²(90°) - 1)
• ± √(sec²(90°) - 1)
• 1/cot 90°

☛ Also check: trigonometry table

What is the Value of Tan 90 Degrees in Terms of Cot 90°?

Since the tangent function is the reciprocal of the cotangent function, we can write tan 90° as 1/cot(90°). The value of cot 90° is equal to 0.

What is the Value of Tan 90° in Terms of Cosec 90°?

Since the tangent function can be represented using the cosecant function, we can write tan 90° as 1/√(cosec²(90°) - 1). The value of cosec 90° is equal to 1.