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Tan 80 Degrees

The value of tan 80 degrees is 5.6712818. . .. Tan 80 degrees in radians is written as tan (80° × π/180°), i.e., tan (4π/9) or tan (1.396263. . .). In this article, we will discuss the methods to find the value of tan 80 degrees with examples.

Tan 80° in decimal: 5.6712818. . .
Tan (-80 degrees): -5.6712818. . .
Tan 80° in radians: tan (4π/9) or tan (1.3962634 . . .)

What is the Value of Tan 80 Degrees?

The value of tan 80 degrees in decimal is 5.671281819. . .. Tan 80 degrees can also be expressed using the equivalent of the given angle (80 degrees) in radians (1.39626 . . .)

We know, using degree to radian conversion, θ in radians = θ in degrees × (pi/180°)
⇒ 80 degrees = 80° × (π/180°) rad = 4π/9 or 1.3962 . . .
∴ tan 80° = tan(1.3962) = 5.6712818. . .

Explanation:

For tan 80 degrees, the angle 80° lies between 0° and 90° (First Quadrant). Since tangent function is positive in the first quadrant, thus tan 80° value = 5.6712818. . .
Since the tangent function is a periodic function, we can represent tan 80° as, tan 80 degrees = tan(80° + n × 180°), n ∈ Z.
⇒ tan 80° = tan 260° = tan 440°, and so on.
Note: Since, tangent is an odd function, the value of tan(-80°) = -tan(80°).

Methods to Find Value of Tan 80 Degrees

The tangent function is positive in the 1st quadrant. The value of tan 80° is given as 5.67128. . .. We can find the value of tan 80 degrees by:

Using Unit Circle
Using Trigonometric Functions

Tan 80 Degrees Using Unit Circle

To find the value of tan 80 degrees using the unit circle:

Rotate ‘r’ anticlockwise to form 80° angle with the positive x-axis.
The tan of 80 degrees equals the y-coordinate(0.9848) divided by x-coordinate(0.1736) of the point of intersection (0.1736, 0.9848) of unit circle and r.

Hence the value of tan 80° = y/x = 5.6713 (approx).

Tan 80° in Terms of Trigonometric Functions

Using trigonometry formulas, we can represent the tan 80 degrees as:

sin(80°)/cos(80°)
± sin 80°/√(1 - sin²(80°))
± √(1 - cos²(80°))/cos 80°
± 1/√(cosec²(80°) - 1)
± √(sec²(80°) - 1)
1/cot 80°

Note: Since 80° lies in the 1st Quadrant, the final value of tan 80° will be positive.

We can use trigonometric identities to represent tan 80° as,

cot(90° - 80°) = cot 10°
-cot(90° + 80°) = -cot 170°
-tan (180° - 80°) = -tan 100°

☛ Also Check:

Examples Using Tan 80 Degrees

Example 1: Find the value of 2 tan(80°)/4 tan(100°).

Solution:

Using trigonometric identities, we know, tan(80°) = -tan(180° - 80°) = -tan 100°.
⇒ tan(80°) = -tan(100°)
⇒ Value of 2 tan(80°)/4 tan(100°) = -2/4 = -1/2
Example 2: Using the value of tan 80°, solve: (sec²(80°) - 1).

Solution:
We know, (sec²(80°) - 1) = (tan²(80°)) = 32.1634
⇒ (sec²(80°) - 1) = 32.1634
Example 3: Find the value of (2 sin (40°) cos (40°) sec (80°)). [Hint: Use tan 80° = 5.6713]

Solution:

Using sin 2a formula,
2 sin (40°) cos (40°) = sin (2 × 40°) = sin 80°
⇒ 2 sin (40°) cos (40°) sec(80°) = sin 80° sec 80°
= sin 80°/cos 80° = tan 80°
⇒ (2 sin (40°) cos (40°) sec(80°)) = 5.6713

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FAQs on Tan 80 Degrees

What is Tan 80 Degrees?

Tan 80 degrees is the value of tangent trigonometric function for an angle equal to 80 degrees. The value of tan 80° is 5.6713 (approx).

How to Find Tan 80° in Terms of Other Trigonometric Functions?

Using trigonometry formula, the value of tan 80° can be given in terms of other trigonometric functions as:

sin(80°)/cos(80°)
± sin 80°/√(1 - sin²(80°))
± √(1 - cos²(80°))/cos 80°
± 1/√(cosec²(80°) - 1)
± √(sec²(80°) - 1)
1/cot 80°

☛ Also check: trigonometry table

What is the Value of Tan 80° in Terms of Cosec 80°?

Since the tangent function can be represented using the cosecant function, we can write tan 80° as 1/√(cosec²(80°) - 1). The value of cosec 80° is equal to 1.01542.

How to Find the Value of Tan 80 Degrees?

The value of tan 80 degrees can be calculated by constructing an angle of 80° with the x-axis, and then finding the coordinates of the corresponding point (0.1736, 0.9848) on the unit circle. The value of tan 80° is equal to the y-coordinate(0.9848) divided by the x-coordinate (0.1736). ∴ tan 80° = 5.6713

What is the Value of Tan 80 Degrees in Terms of Cos 80°?

We know, using trig identities, we can write tan 80° as √(1 - cos²(80°))/cos 80°. Here, the value of cos 80° is equal to 0.173648.

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