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 xaxis.
 The tan of 5pi/12 equals the ycoordinate(0.9659) divided by the xcoordinate(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:
Examples Using Tan 5pi/12

Example 1: Find the value of (2 sin (5pi/24) cos (5pi/24) sec (5pi/12)). [Hint: Use tan 5pi/12 = 3.7321]
Solution:
Using sin 2a formula,
2 sin (5pi/24) cos (5pi/24) = sin (2 × 5pi/24) = sin 5pi/12
⇒ 2 sin (5pi/24) cos (5pi/24) sec(5pi/12) = sin(5pi/12) sec(5pi/12)
= sin(5pi/12)/cos(5pi/12) = tan 5pi/12
⇒ (2 sin (5pi/24) cos (5pi/24) sec(5pi/12)) = 3.7321 
Example 2: Find the value of tan 5pi/12 if cot 5pi/12 is 0.2679.
Solution:
Since, tan 5pi/12 = 1/cot(5pi/12)
⇒ tan 5pi/12 = 1/0.2679 = 3.7321 
Example 3: Find the value of 6 tan(5pi/12)/10 tan(7pi/12).
Solution:
Using trigonometric identities, we know, tan(5pi/12) = tan(pi  5pi/12) = tan 7pi/12.
⇒ tan(5pi/12) = tan(7pi/12)
⇒ Value of 6 tan(5pi/12)/10 tan(7pi/12) = 6/10 = 3/5
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 xaxis, 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 ycoordinate(0.9659) divided by the xcoordinate (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.
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