Tan 5pi/12
The value of tan 5pi/12 is 3.7320508. . .. Tan 5pi/12 radians in degrees is written as tan ((5π/12) × 180°/π), i.e., tan (75°). In this article, we will discuss the methods to find the value of tan 5pi/12 with examples.
- Tan 5pi/12: 2 + √3
- Tan 5pi/12 in decimal: 3.7320508. . .
- Tan (-5pi/12): -3.7320508. . . or -2 - √3
- Tan 5pi/12 in degrees: tan (75°)
What is the Value of Tan 5pi/12?
The value of tan 5pi/12 in decimal is 3.732050807. . .. Tan 5pi/12 can also be expressed using the equivalent of the given angle (5pi/12) in degrees (75°).
We know, using radian to degree conversion, θ in degrees = θ in radians × (180°/pi)
⇒ 5pi/12 radians = 5pi/12 × (180°/pi) = 75° or 75 degrees
∴ tan 5pi/12 = tan 5π/12 = tan(75°) = 2 + √3 or 3.7320508. . .
Explanation:
For tan 5pi/12, the angle 5pi/12 lies between 0 and pi/2 (First Quadrant). Since tangent function is positive in the first quadrant, thus tan 5pi/12 value = 2 + √3 or 3.7320508. . .
Since the tangent function is a periodic function, we can represent tan 5pi/12 as, tan 5pi/12 = tan(5pi/12 + n × pi), n ∈ Z.
⇒ tan 5pi/12 = tan 17pi/12 = tan 29pi/12 , and so on.
Note: Since, tangent is an odd function, the value of tan(-5pi/12) = -tan(5pi/12).
Methods to Find Value of Tan 5pi/12
The tangent function is positive in the 1st quadrant. The value of tan 5pi/12 is given as 3.73205. . .. We can find the value of tan 5pi/12 by:
- Using Trigonometric Functions
- Using Unit Circle
Tan 5pi/12 in Terms of Trigonometric Functions
Using trigonometry formulas, we can represent the tan 5pi/12 as:
- sin(5pi/12)/cos(5pi/12)
- ± sin(5pi/12)/√(1 - sin²(5pi/12))
- ± √(1 - cos²(5pi/12))/cos(5pi/12)
- ± 1/√(cosec²(5pi/12) - 1)
- ± √(sec²(5pi/12) - 1)
- 1/cot(5pi/12)
Note: Since 5pi/12 lies in the 1st Quadrant, the final value of tan 5pi/12 will be positive.
We can use trigonometric identities to represent tan 5pi/12 as,
- cot(pi/2 - 5pi/12) = cot pi/12
- -cot(pi/2 + 5pi/12) = -cot 11pi/12
- -tan (pi - 5pi/12) = -tan 7pi/12
Tan 5pi/12 Using Unit Circle
To find the value of tan 5π/12 using the unit circle:
- Rotate ‘r’ anticlockwise to form 5pi/12 angle with the positive x-axis.
- The tan of 5pi/12 equals the y-coordinate(0.9659) divided by the x-coordinate(0.2588) of the point of intersection (0.2588, 0.9659) of unit circle and r.
Hence the value of tan 5pi/12 = y/x = 3.7321 (approx)
☛ Also Check:
FAQs on Tan 5pi/12
What is Tan 5pi/12?
Tan 5pi/12 is the value of tangent trigonometric function for an angle equal to 5π/12 radians. The value of tan 5pi/12 is 2 + √3 or 3.7321 (approx).
How to Find the Value of Tan 5pi/12?
The value of tan 5pi/12 can be calculated by constructing an angle of 5π/12 radians with the x-axis, and then finding the coordinates of the corresponding point (0.2588, 0.9659) on the unit circle. The value of tan 5pi/12 is equal to the y-coordinate(0.9659) divided by the x-coordinate (0.2588). ∴ tan 5pi/12 = 2 + √3 or 3.7321
What is the Value of Tan 5pi/12 in Terms of Sin 5pi/12?
Using trigonometric identities, we can write tan 5pi/12 in terms of sin 5pi/12 as, tan(5pi/12) = sin(5pi/12)/√(1 - sin²(5pi/12)) . Here, the value of sin 5pi/12 is equal to (√6 + √2)/4.
How to Find Tan 5pi/12 in Terms of Other Trigonometric Functions?
Using trigonometry formula, the value of tan 5pi/12 can be given in terms of other trigonometric functions as:
- sin(5pi/12)/cos(5pi/12)
- ± sin(5pi/12)/√(1 - sin²(5pi/12))
- ± √(1 - cos²(5pi/12))/cos(5pi/12)
- ± 1/√(cosec²(5pi/12) - 1)
- ± √(sec²(5pi/12) - 1)
- 1/cot(5pi/12)
☛ Also check: trigonometry table
What is the Value of Tan 5pi/12 in Terms of Cosec 5pi/12?
Since the tangent function can be represented using the cosecant function, we can write tan 5pi/12 as 1/√(cosec²(5pi/12) - 1). The value of cosec 5pi/12 is equal to 1.03527.