Tan 30 Degrees

In Trigonometry, Sine, Cosine and Tangent are the three primary ratios, based on which the whole trigonometric functions, identities and formulas are designed. Tan 30 degrees along with all trigonometric ratios have their equivalent importance. Basically, these ratios are used to find the angles and the length of sides of a right-angled triangle. These angles are calculated with respect to sin, cos and tan ratios. Usually, the degrees considered are 00, 300, 450, 600 and 900.

The tangent of the angle in a right-angled triangle is equal to the ratio of the opposite side to the adjacent side. Therefore, to find the value of tan 30 degrees, we have to know the value of the opposite side, which is called as perpendicular and adjacent side, which is the base of the triangle.

We can also find out tan 30 degrees value with respect to sin 30 degrees and cos 30 degrees. As we all know, the tangent of an angle is equal to the ratio of sine and cosine of the same angle. If we know the value for sine degrees and cos degrees, then we can also calculate the value for tan degrees.

In this article, we will discuss, the value for tan 30 degrees and also will form a trigonometry table for all values of tan degrees.

Tan 30 Degrees Value

Like Sine and Cosine, Tangent is also a basic function of trigonometry. Most of the trigonometric equation is based on these ratios. Usually, to find the values of sine, cosine and tangent ratios, we use right-angles triangle and also take a unit circle example. First, let us discuss tan 30 degrees value in terms of a right-angled triangle.

tan 30 degrees

Suppose for a triangle ABC, right angled at C, \(\alpha\) is the angle, h is the hypotenuse, b is the adjacent side or base and a is the opposite side or perpendicular. As we know,

Tan\(\alpha\) = \(\frac{Opposite Side}{Adjacent Side}\)

∴ Tan\(\alpha\) = \(\frac{a}{b}\)

Similarly, we can also find the value for Sine and Cosine ratios.

Sin\(\alpha\) = \(\frac{Opposite Side}{Hypotenuse}\)

∴ Sin\(\alpha\) = \(\frac{a}{h}\)

And

Cos\(\alpha\) = \(\frac{Adjacent Side}{Hypotenuse}\)

∴ Cos\(\alpha\) = \(\frac{b}{h}\)

We can also represent the tangent function as the ratio of the sine function and cosine function.

∴ Tan\(\alpha\) = \(\frac{sin \alpha }{cos \alpha }\)

So, tan 30 degrees we can write as;

Tan 300 = \(\frac{sin 30^{\circ} }{cos 30^{\circ} }\)

We know,

Sin 300 = 1/2

&Cos300 = \(\sqrt{3}\)/2

∴ Tan 300 = \(\frac{1/2}{\sqrt{3}/2}\)

Tan 300 = 1/\(\sqrt{3}\)

Hence is the value of Tan 30 degrees.

Unit Circle: For a unit circle also we can calculate the value of tan 30 degrees. Unit circle has a radius as 1 unit and it is drawn on an XY plane. With the below graph, you can check the values of all the trigonometry ratios, such as sin, cos, tan, sec, cot and cosec.

tan 30 degrees

Also, you can see, the values are defined here in terms of radians in case of the unit circle and not in degrees.In the same way, we can derive other values of tangent degrees(00,450,600,900,1800,2700 and 3600). Check the below trigonometry table to get the values for all trigonometry ratios.

Angle

Radian

00

0

300

π/6

450

π/4

600

π/3

900

π/2

1800

π

2700

3π/2

3600

Sin 0 1/2 \(1/\sqrt{2}\) \(\sqrt{3}/2\) 1 0 -1 0
Cos 1 \(\sqrt{3}/2\) \(1/\sqrt{2}\) 1/2 0 -1 0 1
Tan 0 1/\(\sqrt{3}\) 1 \(\sqrt{3}\) 0 0
Cot /\(\sqrt{3}\) 1 1/\(\sqrt{3}\) 0 0
Sec 1 2/\(\sqrt{3}\) /\(\sqrt{2}\) 2 -1 1
Cosec 2 /\(\sqrt{2}\) 2/\(\sqrt{3}\) 1 -1

Trigonometry Formulas Based on Tangent Function

  1. Tan(-\(\Theta\))=-Tan\(\Theta\)
  2. Tan (x+y)= \(\frac{tan x +tan y}{1-tan x tan y}\)
  3. Tan (x-y)=\(\frac{tan x -tan y}{1+tan x tan y}\)
  4. Tan 2x=2 tan x/1-tan2 x
  5. Tan 3x= 3tan x-tan3 x/1-3 tan2 x
  6. Tan (90-\(\Theta\))=Cot\(\Theta\)
  7. Tan (90+\(\Theta\))= -Cot\(\Theta\)

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