If the angle in a right-angled triangle is 30 degrees, then the value of cosine is known as cos of angle 30 degrees. The cosine of angle 30 degrees in the Sexagesimal angle measuring system is written or expressed as cos(30°).
cos(30°) = √3/2
In the fraction format, the value of cos(30°) is equal to √3/2. As it is an irrational number, its value in the decimal form is 0.8660254037…….which is taken as 0.866 approximately in the field of mathematics and for solving the problems. The value of cos(30°) is usually referred to as the function of the standard angle in trigonometry or the trigonometric ratio.
Alternative Form of cos(30°)
In mathematics, there is an alternative format of writing the cos(30°), which is cos(∏/6) (2∏ = 360 degrees. So, ∏ = 180 degrees. ∏/6 = 180°/6 = 30°) in the circular system, also expressed as cos(33 ) in the centesimal system.
The value of cos(∏/6) and cos(33 ) is √3/2 in the fraction form and 0.8660254037……., in the decimal form.
Proof of cos(30°)
In mathematics, to derive the value of cos(30°), there are three ways, namely, two geometrical methods (theoretical approach and practical approach) and one trigonometric method.
Geometrical Method 1 – Theoretical Approach
You must be aware of the direct relationship among the sides of a right-angled triangle when the angle is 30°. As per the properties of a right-angled triangle, if its angle is ∏/6, then the length of the opposite side (not having the angle 30°). Here, PQ) is half the length of the hypotenuse. Based on the same property, the value of cos(30°) is evaluated or assessed theoretically.
Hypotenuse (OP) = d
Perpendicular (PQ) = d/2
Base (OQ) = ?
To find the value of cos(∏/6), it is necessary to find the value of the base.
As it is a right-angled triangle, we can apply the Pythagoras Theorem, and find the value of the length of the adjacent side mathematically.
According to the Pythagoras theorem, in ∆QOP, PQ2 + OQ2 = OP2
(d)2 = (d/2)2 + OQ2
d2 = d2/4 + OQ2
d2 - d2/4 = OQ2
d2 (1 – 1/4) = OQ2
d2 (1 * 4 – 1)/4 = OQ2
d2 (4 – 1)/4 = OQ2
d2 (3/4) = OQ2
OQ = √3d/2
OQ/d = √3/2
In this particular case, d is the length of the hypotenuse, that is, OP.
The value of cosϴ = Base/Hypotenuse (Trigonometric Ratios)
So, OQ/OP = √3/2
Length of the adjacent side or base/Length of the hypotenuse = √3/2
In ∆QOP, the angle is ∏/6, and according to the definition of trigonometric ratio of cosine, the ratio represents cos(30°).
Therefore, the value of cos(30°) is √3/2.
The value of cos(30°) in the fraction form is √3/2 and 0.8660254037….., in the decimal format.
Hence, cos(30°) = √3/2 = 0.8660254037…..
Geometrical Method 2 – Practical Approach
It is possible to calculate the value of cos(∏/6) geometrically by constructing a right-angled triangle with 30 degrees angle and using geometric tools.
Let’s consider the steps mentioned below:
1. Identify a point P on the plane and then horizontally draw a straight line from this point.
2. Take a Dee or protractor and coincide with its centre with point P. Then, proceed by coinciding with its right side base with the straight horizontal line drawn. After that, mark a point at the 30 degrees angle.
3. With the help of a ruler, draw a line from the point P through the 30 degrees angle.
4. From point P, draw an arc on the 30 degrees line by using a compass and opening it of any length. For instance – open the compass of length 7.5cm, and the arc cut the 30 degrees angle at point Q.
5. From the Q, draw a perpendicular to the horizontal line so that it intersects the line at point 0.
By following these five steps, you will have a right-angled triangle ∆QPO, with the 30 degrees angle. Now, we can calculate the value of cos(∏/6) from the triangle ∆QPO mathematically.
cos(30°) = Length of the adjacent side or base/Length of the hypotenuse
cos(30°) = PO/PH
In this particular case, the length of the hypotenuse is 7.5cm, and the length of the adjacent side is 6.5cm (measured with a ruler).
So, cos(30°) = PO/PH = 6.5/7.5
cos(30°) = 0.8660254037…..
By using the cos square identity in trigonometry i.e., cos2ϴ = 1 – sin2 ϴ, we can evaluate the exact value of cos(33 ). For calculating the exact value of cos(∏/6), we have to substitute the value of sin(30°) in the same formula.
cos(30°) = √1 – sin230°
The value of sin30° is 1/2 (Trigonometric Ratios)
cos(30°) = √1 – (1/2)2
cos(30°) = √1 – (1/4)
cos(30°) = √(1 * 4 – 1)/4
cos(30°) = √(4 – 1)/4
cos(30°) = √3/4
Therefore, cos(30°) = √3/2
With the help of both the trigonometric method and the geometrical methods(the theoretical approach and the practical approach), we have proved that the value of cos(∏/6) is √3/2 in the fraction format and 0.8660254037……, in the decimal format, with the approximate value equal to 0.866.Besides, we have also proved that in the practical approach of the geometrical method, the value of cos(30°) is equal to 0.8666666666. Now, you can compare both the values of cos(30°) and observe that the value of cos(30°) obtained in the practical approach differs slightly from the values obtained using the theoretical approach of the geometrical method and the trigonometric method. On the other hand, the approximate value of cos(∏/6) is the same in all cases.