Tan 5pi/3
The value of tan 5pi/3 is -1.7320508. . .. Tan 5pi/3 radians in degrees is written as tan ((5π/3) × 180°/π), i.e., tan (300°). In this article, we will discuss the methods to find the value of tan 5pi/3 with examples.
- Tan 5pi/3: -√3
- Tan 5pi/3 in decimal: -1.7320508. . .
- Tan (-5pi/3): 1.7320508. . . or √3
- Tan 5pi/3 in degrees: tan (300°)
What is the Value of Tan 5pi/3?
The value of tan 5pi/3 in decimal is -1.732050807. . .. Tan 5pi/3 can also be expressed using the equivalent of the given angle (5pi/3) in degrees (300°).
We know, using radian to degree conversion, θ in degrees = θ in radians × (180°/pi)
⇒ 5pi/3 radians = 5pi/3 × (180°/pi) = 300° or 300 degrees
∴ tan 5pi/3 = tan 5π/3 = tan(300°) = -√3 or -1.7320508. . .
Explanation:
For tan 5pi/3, the angle 5pi/3 lies between 3pi/2 and 2pi (Fourth Quadrant). Since tangent function is negative in the fourth quadrant, thus tan 5pi/3 value = -√3 or -1.7320508. . .
Since the tangent function is a periodic function, we can represent tan 5pi/3 as, tan 5pi/3 = tan(5pi/3 + n × pi), n ∈ Z.
⇒ tan 5pi/3 = tan 8pi/3 = tan 11pi/3 , and so on.
Note: Since, tangent is an odd function, the value of tan(-5pi/3) = -tan(5pi/3).
Methods to Find Value of Tan 5pi/3
The tangent function is negative in the 4th quadrant. The value of tan 5pi/3 is given as -1.73205. . .. We can find the value of tan 5pi/3 by:
- Using Trigonometric Functions
- Using Unit Circle
Tan 5pi/3 in Terms of Trigonometric Functions
Using trigonometry formulas, we can represent the tan 5pi/3 as:
- sin(5pi/3)/cos(5pi/3)
- ± sin(5pi/3)/√(1 - sin²(5pi/3))
- ± √(1 - cos²(5pi/3))/cos(5pi/3)
- ± 1/√(cosec²(5pi/3) - 1)
- ± √(sec²(5pi/3) - 1)
- 1/cot(5pi/3)
Note: Since 5pi/3 lies in the 4th Quadrant, the final value of tan 5pi/3 will be negative.
We can use trigonometric identities to represent tan 5pi/3 as,
- cot(pi/2 - 5pi/3) = cot(-7pi/6)
- -cot(pi/2 + 5pi/3) = -cot 13pi/6
- -tan (pi - 5pi/3) = -tan(-2pi/3)
Tan 5pi/3 Using Unit Circle
To find the value of tan 5π/3 using the unit circle:
- Rotate ‘r’ anticlockwise to form 5pi/3 angle with the positive x-axis.
- The tan of 5pi/3 equals the y-coordinate(-0.866) divided by the x-coordinate(0.5) of the point of intersection (0.5, -0.866) of unit circle and r.
Hence the value of tan 5pi/3 = y/x = -1.7321 (approx)
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FAQs on Tan 5pi/3
What is Tan 5pi/3?
Tan 5pi/3 is the value of tangent trigonometric function for an angle equal to 5π/3 radians. The value of tan 5pi/3 is -√3 or -1.7321 (approx).
What is the Value of Tan 5pi/3 in Terms of Cosec 5pi/3?
Since the tangent function can be represented using the cosecant function, we can write tan 5pi/3 as -1/√(cosec²(5pi/3) - 1). The value of cosec 5pi/3 is equal to -1.15470.
How to Find the Value of Tan 5pi/3?
The value of tan 5pi/3 can be calculated by constructing an angle of 5π/3 radians with the x-axis, and then finding the coordinates of the corresponding point (0.5, -0.866) on the unit circle. The value of tan 5pi/3 is equal to the y-coordinate(-0.866) divided by the x-coordinate (0.5). ∴ tan 5pi/3 = -√3 or -1.7321
What is the Value of Tan 5pi/3 in Terms of Sin 5pi/3?
Using trigonometric identities, we can write tan 5pi/3 in terms of sin 5pi/3 as, tan(5pi/3) = sin(5pi/3)/√(1 - sin²(5pi/3)) . Here, the value of sin 5pi/3 is equal to -(√3/2).
How to Find Tan 5pi/3 in Terms of Other Trigonometric Functions?
Using trigonometry formula, the value of tan 5pi/3 can be given in terms of other trigonometric functions as:
- sin(5pi/3)/cos(5pi/3)
- ± sin(5pi/3)/√(1 - sin²(5pi/3))
- ± √(1 - cos²(5pi/3))/cos(5pi/3)
- ± 1/√(cosec²(5pi/3) - 1)
- ± √(sec²(5pi/3) - 1)
- 1/cot(5pi/3)
☛ Also check: trigonometry table