Sec 30 Degrees Exact Value with Proof

Trigonometry is a branch of mathematics which deals with the study of measurements of sides and angles of a three sided geometric figure (triangle). It establishes the relationship between the sides and angles of a right triangle. The two basic trigonometric ratios are sine and cosine of an angle. All the other four trigonometric ratios can be defined with these two basic trigonometric functions. In a right triangle, the side opposite to the right angle is the longest side and is called the hypotenuse. If any of the other two angles except right angle is considered to be the reference angle, then the side that is adjacent to the reference angle is called the adjacent side or the base and the side opposite to the reference angle is called the opposite side or the perpendicular. Secant of an angle is defined as the ratio of hypotenuse and the adjacent side of the reference angle. 

What is Sec 30 Degree Value?

Secant of any angle is described as a quotient obtained by dividing the hypotenuse of the right triangle by the side adjacent to the angle whose secant is to be determined. Secant of any angle is the transpose or the reciprocal of its cosine. So, secant and cosine of an angle can be related as: \[Sec\theta = \frac{1}{Cos\theta}\]. Here θ is the reference angle whose trigonometric ratio is to be determined. Cosine of the angle equal to 300 is equal to \[\frac{\sqrt{3}}{2}\]. So, secant of the angle equal to 30o can be written as \[\frac{2}{\sqrt{3}}\].

How to Find Sec 30 Degrees?

The concept of trigonometric ratios of standard angles help in finding the value of sec 30 degrees. To find the trigonometric ratios of the standard angle 300, an equilateral triangle is constructed with side ‘a’. The altitude of the triangle is drawn to one of its sides from the opposite vertex as shown in the figure below. 

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So, from the above figure, is evident that the altitude AD divides the base BC into two equal halves BD and CD measuring a/2 units and bisects the angle BAC. So, the angle BAD = 300. Considering the right triangle ABD, the Pythagorean theorem states that: AD2 + BD2  = AB2.

Using this relationship, AD can be calculated as √3a/2. Now let us consider the angle BAD as the reference angle. The hypotenuse of the right triangle BAD is AB and the side adjacent to the reference angle BAD is AD. Secant of angle BAD is the ratio of hypotenuse and the adjacent side. The measure of AB is ‘a’ units and the measure of AD is \[\frac{\sqrt{3}a}{2}\] units.

Sec(BAD) = \[\frac{Hypotenuse}{Adjacent/ Base}\]

Sec 30° = \[\frac{a}{\frac{\sqrt{3}}{2}a}\]

Sec 30° = \[\frac{2a}{\sqrt{3}a}\]

Sec 30° = \[\frac{2}{\sqrt{3}}\]

Example Problems:

Question 1) Find the value of Cos 300 . Sec 300 - Sec2 300.

Solution:

Cos 300 . Sec 300 - Sec2 300 = \[\frac{\sqrt{3}}{2}\times \frac{2}{\sqrt{3}} - (\frac{\sqrt{3}}{2})^{2}\]

= 1 - \[\frac{3}{4}\]

= \[\frac{1}{4}\]

Question 2) Find the value of Sec-1 (Sec 300) and Sec (Sec-1 √3/2)

Solution:

Sec 30 degree value = \[\frac{\sqrt{3}}{2}\]

Secant and inverse secant are inverse operations with respect to each other.

So, Sec-1 (Sec 300) = 300

Sec (Sec-1 \[\frac{\sqrt{3}}{2}\]) = \[\frac{\sqrt{3}}{2}\]

Fun Facts:

• The English statement “Some People Have Curly Brown Hairs Turned Permanently Black” can be used to remember the definitions of the three basic trigonometric ratios.

• Secant of a reference angle is the reciprocal of its cosine. It should never be confused with the inverse of its cosine. Inverse trigonometric functions are entirely different.