Tangent of 30 Degrees
The value of the tangent of 30 degrees is 0.5773502. . .. Tan 30 degrees in radians is written as tan (30° × π/180°), i.e., tan (π/6) or tan (0.523598. . .). In this article, we will discuss the methods to find the value of tan 30 degrees with examples.
- Tangent of 30 as a fraction: 1/√3 (or) √3/3
- Tan 30° in decimal: 0.5773502. . .
- Tan (-30 degrees): -0.5773502. . . or -1/√3
- Tan 30° in radians: tan (π/6) or tan (0.5235987 . . .)
What is the Value of Tan 30 Degrees?
The value of tan 30 degrees in decimal is 0.577350269.... The tangent of 30 degrees can be found by taking the sine of 30 degrees and dividing it by the cosine of 30 degrees. Since the sine of 30 degrees is 1/2 and the cosine of 30 degrees is √3/2, the tangent of 30 degrees is
tan 30° = (sin 30°)/(cos 30°) = (1/2) / (√3/2) = 1/√3
By rationalizing the denominator, the tangent of 30 can also be written as √3/3. Its decimal equivalent is 0.577350269.
Tangent of 30 in Radians
Tan 30 degrees can also be expressed using the equivalent of the given angle (30 degrees) in radians (0.52359 . . .)
We know, using degree to radian conversion, θ in radians = θ in degrees × (pi/180°)
⇒ 30 degrees = 30° × (π/180°) rad = π/6 or 0.5235 . . .
∴ tan 30° = tan(0.5235) = 1/√3 or 0.5773502. . .
Explanation:
For tan 30 degrees, the angle 30° lies between 0° and 90° (First Quadrant). Since tangent function is positive in the first quadrant, thus tan 30° value = 1/√3 or 0.5773502. . .
Since the tangent function is a periodic function, we can represent tan 30° as, tan 30 degrees = tan(30° + n × 180°), n ∈ Z.
⇒ tan 30° = tan 210° = tan 390°, and so on.
Note: Since, tangent is an odd function, the value of tan(-30°) = -tan(30°).
Finding Tangent of 30 Degrees
The tangent function is positive in the 1st quadrant. The value of tan 30° is given as 0.57735.... We can find the value of tan 30 degrees by:
- Using Calculator
- Using Trigonometric Identity
- Using Unit Circle
- Using Special Triangles
Tan 30 Degrees by Calculator
It is very easy to find the value of tan 30° by a scientific or graphing calculator. Simply set the calculator in "degrees mode" and type tan (30), then it will show its value as 0.5773502...
Tangent of 30 Value by Trigonometric identity
We have a trigonometric identity that says tan x = (sin x)/ (cos x). By substituting x = 30° here, we get tan 30° = (sin 30°)/(cos 30°) = (1/2) / (√3/2) = 1/2 × 2/√3 = 1/√3 (or) √3/3.
Tan 30 Degree Value by Unit Circle
o find the value of tan 30 degrees using the unit circle:
- Rotate ‘r’ anticlockwise to form 30° angle with the positive x-axis.
- The tan of 30 degrees equals the y-coordinate(0.5) divided by the x-coordinate(0.866) of the point of intersection (0.866, 0.5) of the unit circle and r.
Hence the value of tan 30° = y/x = 0.5774 (approx).
Value of Tan 30 Using Special Triangles
Another method to find the tangent of 30 degrees is to use the special right triangle, which is a right triangle with angles of 30 degrees, 60 degrees, and 90 degrees which is as follows:
By the definition of tangent function:
tan 30° = x / (x√3) = 1/√3 (or) √3/3.
Tan 30° in Terms of Trigonometric Functions
Using trigonometry formulas, we can represent the tan 30 degrees as:
- sin(30°)/cos(30°)
- ± sin 30°/√(1 - sin²(30°))
- ± √(1 - cos²(30°))/cos 30°
- ± 1/√(cosec²(30°) - 1)
- ± √(sec²(30°) - 1)
- 1/cot 30°
Note: Since 30° lies in the 1st Quadrant, the final value of tan 30° will be positive.
We can use trigonometric identities to represent tan 30° as,
- cot(90° - 30°) = cot 60°
- -cot(90° + 30°) = -cot 120°
- -tan (180° - 30°) = -tan 150°
T☛ Related Articles:
FAQs on Tan 30 Degrees
What is Tangent of 30 Degrees?
Tangent of 30 degrees is the value of tangent trigonometric function for an angle equal to 30 degrees. The value of tan 30° is 1/√3 or 0.5774 (approx). In fraction form, the tangent of 30 is 1/√3 (or) √3/3.
What is the Value of Tan 30 Degrees as a Fraction?
The value of tan 30 degrees as a fraction is 1/√3. But we can rationalize the denominator by multiplying and dividing by √3. Then we get tan 30° = 1/√3 × √3/√3 = √3/3. Thus, the value of tan 30 degrees in fraction form can be either 1/√3 or √3/3.
How to Find the Value of Tan 30 Degrees?
The value of tan 30 degrees can be calculated by constructing an angle of 30° 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 30° is equal to the y-coordinate(0.5) divided by the x-coordinate (0.866). ∴ tan 30° = 1/√3 or 0.5774
How to Find Tan 30° in Terms of Other Trigonometric Functions?
Using trigonometry formula, the value of tan 30° can be given in terms of other trigonometric functions as:
- Since we have tan x = (sin x)/ cos x), tan 30° = sin(30°)/cos(30°)
- Now, we have cos x = ± √(1 - sin²x), tan 30° = ± sin 30°/√(1 - sin²(30°))
- In step 1, we can use sin x = ± √(1 - cos²x), then tan 30° = ± √(1 - cos²(30°))/cos 30°
- By identity tan2x = sec2x - 1, we have tan 30° = ± √(sec²(30°) - 1)
- By identity, tan x = 1/cot x, we can write tan 30° = 1/cot 30°
☛ Also check: trigonometry table
What is the Value of Tan 30° in Terms of Sec 30°?
We can represent the tangent function in terms of the secant function using trig identities, tan 30° can be written as √(sec²(30°) - 1). Here, the value of sec 30° is equal to 1.1547.
What is the Value of Tan 30 Degrees in Terms of Sin 30°?
Using trigonometric identities, we can write tan 30° in terms of sin 30° as, tan(30°) = sin 30°/√(1 - sin²(30°)) . Here, the value of sin 30° is equal to 1/2.