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Tan 365 Degrees

The value of tan 365 degrees is 0.0874886. . .. Tan 365 degrees in radians is written as tan (365° × π/180°), i.e., tan (73π/36) or tan (6.370451. . .). In this article, we will discuss the methods to find the value of tan 365 degrees with examples.

Tan 365° in decimal: 0.0874886. . .
Tan (-365 degrees): -0.0874886. . .
Tan 365° in radians: tan (73π/36) or tan (6.3704517 . . .)

What is the Value of Tan 365 Degrees?

The value of tan 365 degrees in decimal is 0.087488663. . .. Tan 365 degrees can also be expressed using the equivalent of the given angle (365 degrees) in radians (6.37045 . . .)

We know, using degree to radian conversion, θ in radians = θ in degrees × (pi/180°)
⇒ 365 degrees = 365° × (π/180°) rad = 73π/36 or 6.3704 . . .
∴ tan 365° = tan(6.3704) = 0.0874886. . .

Explanation:

For tan 365°, the angle 365° > 360°. We can represent tan 365° as, tan(365° mod 360°) = tan(5°). The angle 365°, coterminal to angle 5°, is located in the First Quadrant(Quadrant I).
Since tangent function is positive in the 1st quadrant, thus tan 365 degrees value = 0.0874886. . .
Similarly, given the periodic property of tan 365°, it can also be written as, tan 365 degrees = (365° + n × 180°), n ∈ Z.
⇒ tan 365° = tan 545° = tan 725°, and so on.
Note: Since, tangent is an odd function, the value of tan(-365°) = -tan(365°).

Methods to Find Value of Tan 365 Degrees

The tangent function is positive in the 1st quadrant. The value of tan 365° is given as 0.08748. . .. We can find the value of tan 365 degrees by:

Using Trigonometric Functions
Using Unit Circle

Tan 365° in Terms of Trigonometric Functions

Using trigonometry formulas, we can represent the tan 365 degrees as:

sin(365°)/cos(365°)
± sin 365°/√(1 - sin²(365°))
± √(1 - cos²(365°))/cos 365°
± 1/√(cosec²(365°) - 1)
± √(sec²(365°) - 1)
1/cot 365°

Note: Since 365° lies in the 1st Quadrant, the final value of tan 365° will be positive.

We can use trigonometric identities to represent tan 365° as,

cot(90° - 365°) = cot(-275°)
-cot(90° + 365°) = -cot 455°
-tan (180° - 365°) = -tan(-185°)

Tan 365 Degrees Using Unit Circle

To find the value of tan 365 degrees using the unit circle, represent 365° in the form (1 × 360°) + 5° [∵ 365°>360°] ∵ The angle 365° is coterminal to 5° angle and also tangent is a periodic function, tan 365° = tan 5°.

Rotate ‘r’ anticlockwise to form 5° or 365° angle with the positive x-axis.
The tan of 365 degrees equals the y-coordinate(0.0872) divided by x-coordinate(0.9962) of the point of intersection (0.9962, 0.0872) of unit circle and r.

Hence the value of tan 365° = y/x = 0.0875 (approx).

☛ Also Check:

Examples Using Tan 365 Degrees

Example 1: Find the value of (2 sin (182.5°) cos (182.5°) sec (365°)). [Hint: Use tan 365° = 0.0875]

Solution:

Using sin 2a formula,
2 sin (182.5°) cos (182.5°) = sin (2 × 182.5°) = sin 365°
⇒ 2 sin (182.5°) cos (182.5°) sec(365°) = sin 365° sec 365°
= sin 365°/cos 365° = tan 365°
⇒ (2 sin (182.5°) cos (182.5°) sec(365°)) = 0.0875
Example 2: Find the value of 5 tan(365°)/7 tan(-185°).

Solution:

Using trigonometric identities, we know, tan(365°) = -tan(180° - 365°) = -tan(-185°).
⇒ tan(365°) = -tan(-185°)
⇒ Value of 5 tan(365°)/7 tan(-185°) = -5/7
Example 3: Using the value of tan 365°, solve: (sec²(365°) - 1).

Solution:
We know, (sec²(365°) - 1) = (tan²(365°)) = 0.0077
⇒ (sec²(365°) - 1) = 0.0077

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FAQs on Tan 365 Degrees

What is Tan 365 Degrees?

Tan 365 degrees is the value of tangent trigonometric function for an angle equal to 365 degrees. The value of tan 365° is 0.0875 (approx).

What is the Value of Tan 365° in Terms of Cosec 365°?

Since the tangent function can be represented using the cosecant function, we can write tan 365° as 1/√(cosec²(365°) - 1). The value of cosec 365° is equal to 11.47371.

How to Find the Value of Tan 365 Degrees?

The value of tan 365 degrees can be calculated by constructing an angle of 365° with the x-axis, and then finding the coordinates of the corresponding point (0.9962, 0.0872) on the unit circle. The value of tan 365° is equal to the y-coordinate(0.0872) divided by the x-coordinate (0.9962). ∴ tan 365° = 0.0875

What is the Value of Tan 365 Degrees in Terms of Cos 365°?

We know, using trig identities, we can write tan 365° as √(1 - cos²(365°))/cos 365°. Here, the value of cos 365° is equal to 0.996194.

How to Find Tan 365° in Terms of Other Trigonometric Functions?

Using trigonometry formula, the value of tan 365° can be given in terms of other trigonometric functions as:

sin(365°)/cos(365°)
± sin 365°/√(1 - sin²(365°))
± √(1 - cos²(365°))/cos 365°
± 1/√(cosec²(365°) - 1)
± √(sec²(365°) - 1)
1/cot 365°

☛ Also check: trigonometric table

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