Tan 5pi/6
The value of tan 5pi/6 is -0.5773502. . .. Tan 5pi/6 radians in degrees is written as tan ((5π/6) × 180°/π), i.e., tan (150°). In this article, we will discuss the methods to find the value of tan 5pi/6 with examples.
- Tan 5pi/6: -1/√3
- Tan 5pi/6 in decimal: -0.5773502. . .
- Tan (-5pi/6): 0.5773502. . . or 1/√3
- Tan 5pi/6 in degrees: tan (150°)
What is the Value of Tan 5pi/6?
The value of tan 5pi/6 in decimal is -0.577350269. . .. Tan 5pi/6 can also be expressed using the equivalent of the given angle (5pi/6) in degrees (150°).
We know, using radian to degree conversion, θ in degrees = θ in radians × (180°/pi)
⇒ 5pi/6 radians = 5pi/6 × (180°/pi) = 150° or 150 degrees
∴ tan 5pi/6 = tan 5π/6 = tan(150°) = -1/√3 or -0.5773502. . .
Explanation:
For tan 5pi/6, the angle 5pi/6 lies between pi/2 and pi (Second Quadrant). Since tangent function is negative in the second quadrant, thus tan 5pi/6 value = -1/√3 or -0.5773502. . .
Since the tangent function is a periodic function, we can represent tan 5pi/6 as, tan 5pi/6 = tan(5pi/6 + n × pi), n ∈ Z.
⇒ tan 5pi/6 = tan 11pi/6 = tan 17pi/6 , and so on.
Note: Since, tangent is an odd function, the value of tan(-5pi/6) = -tan(5pi/6).
Methods to Find Value of Tan 5pi/6
The tangent function is negative in the 2nd quadrant. The value of tan 5pi/6 is given as -0.57735. . .. We can find the value of tan 5pi/6 by:
- Using Unit Circle
- Using Trigonometric Functions
Tan 5pi/6 Using Unit Circle
To find the value of tan 5π/6 using the unit circle:
- Rotate ‘r’ anticlockwise to form 5pi/6 angle with the positive x-axis.
- The tan of 5pi/6 equals the y-coordinate(0.5) divided by the x-coordinate(-0.866) of the point of intersection (-0.866, 0.5) of unit circle and r.
Hence the value of tan 5pi/6 = y/x = -0.5774 (approx)
Tan 5pi/6 in Terms of Trigonometric Functions
Using trigonometry formulas, we can represent the tan 5pi/6 as:
- sin(5pi/6)/cos(5pi/6)
- ± sin(5pi/6)/√(1 - sin²(5pi/6))
- ± √(1 - cos²(5pi/6))/cos(5pi/6)
- ± 1/√(cosec²(5pi/6) - 1)
- ± √(sec²(5pi/6) - 1)
- 1/cot(5pi/6)
Note: Since 5pi/6 lies in the 2nd Quadrant, the final value of tan 5pi/6 will be negative.
We can use trigonometric identities to represent tan 5pi/6 as,
- cot(pi/2 - 5pi/6) = cot(-pi/3)
- -cot(pi/2 + 5pi/6) = -cot 4pi/3
- -tan (pi - 5pi/6) = -tan pi/6
☛ Also Check:
FAQs on Tan 5pi/6
What is Tan 5pi/6?
Tan 5pi/6 is the value of tangent trigonometric function for an angle equal to 5π/6 radians. The value of tan 5pi/6 is -1/√3 or -0.5774 (approx).
How to Find the Value of Tan 5pi/6?
The value of tan 5pi/6 can be calculated by constructing an angle of 5π/6 radians with the x-axis, and then finding the coordinates of the corresponding point (-0.866, 0.5) on the unit circle. The value of tan 5pi/6 is equal to the y-coordinate(0.5) divided by the x-coordinate (-0.866). ∴ tan 5pi/6 = -1/√3 or -0.5774
What is the Value of Tan 5pi/6 in Terms of Sec 5pi/6?
We can represent the tangent function in terms of the secant function using trig identities, tan 5pi/6 can be written as -√(sec²(5pi/6) - 1). Here, the value of sec 5pi/6 is equal to -1.1547.
What is the Value of Tan 5pi/6 in Terms of Sin 5pi/6?
Using trigonometric identities, we can write tan 5pi/6 in terms of sin 5pi/6 as, tan(5pi/6) = -sin(5pi/6)/√(1 - sin²(5pi/6)) . Here, the value of sin 5pi/6 is equal to 1/2.
How to Find Tan 5pi/6 in Terms of Other Trigonometric Functions?
Using trigonometry formula, the value of tan 5pi/6 can be given in terms of other trigonometric functions as:
- sin(5pi/6)/cos(5pi/6)
- ± sin(5pi/6)/√(1 - sin²(5pi/6))
- ± √(1 - cos²(5pi/6))/cos(5pi/6)
- ± 1/√(cosec²(5pi/6) - 1)
- ± √(sec²(5pi/6) - 1)
- 1/cot(5pi/6)
☛ Also check: trigonometry table