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Tan 78 Degrees

The value of tan 78 degrees is 4.7046301. . .. Tan 78 degrees in radians is written as tan (78° × π/180°), i.e., tan (13π/30) or tan (1.361356. . .). In this article, we will discuss the methods to find the value of tan 78 degrees with examples.

Tan 78° in decimal: 4.7046301. . .
Tan (-78 degrees): -4.7046301. . .
Tan 78° in radians: tan (13π/30) or tan (1.3613568 . . .)

What is the Value of Tan 78 Degrees?

The value of tan 78 degrees in decimal is 4.704630109. . .. Tan 78 degrees can also be expressed using the equivalent of the given angle (78 degrees) in radians (1.36135 . . .)

We know, using degree to radian conversion, θ in radians = θ in degrees × (pi/180°)
⇒ 78 degrees = 78° × (π/180°) rad = 13π/30 or 1.3613 . . .
∴ tan 78° = tan(1.3613) = 4.7046301. . .

Explanation:

For tan 78 degrees, the angle 78° lies between 0° and 90° (First Quadrant). Since tangent function is positive in the first quadrant, thus tan 78° value = 4.7046301. . .
Since the tangent function is a periodic function, we can represent tan 78° as, tan 78 degrees = tan(78° + n × 180°), n ∈ Z.
⇒ tan 78° = tan 258° = tan 438°, and so on.
Note: Since, tangent is an odd function, the value of tan(-78°) = -tan(78°).

Methods to Find Value of Tan 78 Degrees

The tangent function is positive in the 1st quadrant. The value of tan 78° is given as 4.70463. . .. We can find the value of tan 78 degrees by:

Using Trigonometric Functions
Using Unit Circle

Tan 78° in Terms of Trigonometric Functions

Using trigonometry formulas, we can represent the tan 78 degrees as:

sin(78°)/cos(78°)
± sin 78°/√(1 - sin²(78°))
± √(1 - cos²(78°))/cos 78°
± 1/√(cosec²(78°) - 1)
± √(sec²(78°) - 1)
1/cot 78°

Note: Since 78° lies in the 1st Quadrant, the final value of tan 78° will be positive.

We can use trigonometric identities to represent tan 78° as,

cot(90° - 78°) = cot 12°
-cot(90° + 78°) = -cot 168°
-tan (180° - 78°) = -tan 102°

Tan 78 Degrees Using Unit Circle

To find the value of tan 78 degrees using the unit circle:

Rotate ‘r’ anticlockwise to form 78° angle with the positive x-axis.
The tan of 78 degrees equals the y-coordinate(0.9781) divided by x-coordinate(0.2079) of the point of intersection (0.2079, 0.9781) of unit circle and r.

Hence the value of tan 78° = y/x = 4.7046 (approx).

☛ Also Check:

Examples Using Tan 78 Degrees

Example 1: Using the value of tan 78°, solve: (sec²(78°) - 1).

Solution:
We know, (sec²(78°) - 1) = (tan²(78°)) = 22.1335
⇒ (sec²(78°) - 1) = 22.1335
Example 2: Find the value of (2 sin (39°) cos (39°) sec (78°)). [Hint: Use tan 78° = 4.7046]

Solution:

Using sin 2a formula,
2 sin (39°) cos (39°) = sin (2 × 39°) = sin 78°
⇒ 2 sin (39°) cos (39°) sec(78°) = sin 78° sec 78°
= sin 78°/cos 78° = tan 78°
⇒ (2 sin (39°) cos (39°) sec(78°)) = 4.7046
Example 3: Simplify: 3 (tan 78°/cot 12°)

Solution:

We know tan 78° = cot 12°
⇒ 3 tan 78°/cot 12° = 3 (tan 78°/tan 78°)
= 3(1) = 3

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FAQs on Tan 78 Degrees

What is Tan 78 Degrees?

Tan 78 degrees is the value of tangent trigonometric function for an angle equal to 78 degrees. The value of tan 78° is 4.7046 (approx).

How to Find the Value of Tan 78 Degrees?

The value of tan 78 degrees can be calculated by constructing an angle of 78° with the x-axis, and then finding the coordinates of the corresponding point (0.2079, 0.9781) on the unit circle. The value of tan 78° is equal to the y-coordinate(0.9781) divided by the x-coordinate (0.2079). ∴ tan 78° = 4.7046

What is the Value of Tan 78 Degrees in Terms of Cos 78°?

We know, using trig identities, we can write tan 78° as √(1 - cos²(78°))/cos 78°. Here, the value of cos 78° is equal to 0.207911.

How to Find Tan 78° in Terms of Other Trigonometric Functions?

Using trigonometry formula, the value of tan 78° can be given in terms of other trigonometric functions as:

sin(78°)/cos(78°)
± sin 78°/√(1 - sin²(78°))
± √(1 - cos²(78°))/cos 78°
± 1/√(cosec²(78°) - 1)
± √(sec²(78°) - 1)
1/cot 78°

☛ Also check: trigonometry table

What is the Value of Tan 78° in Terms of Cosec 78°?

Since the tangent function can be represented using the cosecant function, we can write tan 78° as 1/√(cosec²(78°) - 1). The value of cosec 78° is equal to 1.02234.

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