# Sec 30

In trigonometry, we have learned the three main primary functions, such as sine, cosine and tangent along with them the other three trigonometric functions, such as secant, cotangent and cosecant. Here, we will find the value of sec 30 degrees along with the other secant degree values.

Secant function is basically the transpose of the cosine function. Therefore, to find the value of sec 30, we have to find the value of cos 30. We take the reference of a right angle triangle to get the values of trigonometric functions or ratios such as sin, cos, tan, sec, cot, csc.

As we know, in a right angle triangle, Cosine function defines a relation between the adjacent side and the hypotenuse of a right-angled triangle with respect to the angle, formed between the adjacent side and the hypotenuse. Therefore, we can say the secant function defines a relation between the hypotenuse and adjacent side of the right-angled triangle with respect to the angle, formed between the adjacent side and the hypotenuse. Hence sec 30 value will be the ratio of the hypotenuse and adjacent side.

We will derive the value of sec 30 degrees with the help of cos function, along with the values of other degrees such as 00, 450, 600, 900,1800 which are generally used in trigonometry equations. Let us discuss here.

## Sec 30 Value

We know that, in a right-angled triangle, the secant of ∠α is a ratio of the length of the hypotenuse and the adjacent side to the angle, where ∠α is the angle formed between the adjacent side and the hypotenuse.

secant ∠α = $\frac{Hypotenuse}{Adjacent Side}$

= $\frac{Hypotenuse}{Base}$

sec ∠α = $\frac{h}{b}$

Now, we know, sec ∠α = $\frac{1}{cos ∠α}$

Therefore, sec 300 = $\frac{1}{cos 30^{\circ}}$

The value of cos 300 = $\frac{\sqrt{3}}{2}$

Hence, the value of sec 300 = $\frac{2}{\sqrt{3}}$

In the same way, we can write the other values sec ratios, such as,

sec 00 = $\frac{1}{cos 0^{\circ}}$ = 1/1 = 1

sec 450 = $\frac{1}{cos 45^{\circ}}$ = $\frac{1}{\frac{1}{\sqrt{2}}}$ = $\sqrt{2}$

sec 600 = $\frac{1}{cos 60^{\circ}}$ = $\frac{1}{\frac{1}{2}}$ = 2

sec 900 = $\frac{1}{cos 90^{\circ}}$ = $\frac{1}{0}$ = ∞

Given below is the trigonometry table for the trigonometric ratios covering the values of degrees 00, 300, 450, 600, 900.

 00 300 450 600 900 Sin 0 $\frac{1}{2}$ $\frac{1}{\sqrt{2}}$ $\frac{\sqrt{3}}{2}$ 1 Cos 1 $\frac{\sqrt{3}}{2}$ $\frac{1}{\sqrt{2}}$ $\frac{1}{2}$ 0 Tan 0 $\frac{1}{\sqrt{3}}$ 1 $\sqrt{3}$ ∞ Sec 1 $\frac{2}{\sqrt{3}}$ $\sqrt{2}$ 2 ∞ Csc ∞ 2 $\sqrt{2}$ $\frac{2}{\sqrt{3}}$ 1 Cot ∞ $\sqrt{3}$ 1 $\frac{1}{\sqrt{3}}$ 0

### Sec 30 Example Problem

Example: compute 2 sec 300 + 2 sec 600.

Solution:

Given,2 sec 300 + 2 sec 600

We know, sec 300 = $\frac{2}{\sqrt{3}}$ and cos 600 = 1/2

Therefore, 2 × $\frac{2}{\sqrt{3}}$ + 2 × ½

= $\frac{4}{\sqrt{3}}$ + 1

2 sec 300 + 2 sec 600= $\frac{4+\sqrt{3}}{3}$.