The cosine function in trigonometry is calculated by taking the ratio of the adjacent side to the hypotenuse of the triangle. Say the angle of a right angle triangle is at 30 degrees, so the value of the cosine at this particular angle is the division of 0.8660254037 The value of sec 30 will be the exact reciprocal of the value of cos 30.
[cos(30^{o}) = frac{sqrt{3}}{2}]
In the fraction format, the value of cos(30°) is equal to 0.8660254037. 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.
The 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 circulatory system, also expressed as cos(33 ) in the centesimal system. The value of cos(∏/6) and cos(33 ) is [frac{sqrt{3}}{2}] in the fraction form and 0.8660254037……., in the decimal form.
Proof for Cos 30 Degrees
It is always important in mathematics to be able to properly derive a value using a number of approaches in order to take it for granted. The value of cos 30 can also be found out using the theoretical and practical approaches. These are the geometric methods of finding the value of cos 30, and there is also one trigonometric method which can be employed.
Using the Theoretical Approach
For any right angle triangle, a direct and established relationship exists among the sides if the angle is at 30 degrees. The length of the opposite side of the triangle is always half of the length of the hypotenuse. Hence, we know the value of two sides, the hypotenuse and the opposite side to the hypotenuse, so we can try to employ the Pythagorean theorem to find the value of the adjacent side.
Using the Practical Approach
There is another way to find out the value of cos 30, this is using the practical approach. Follow these steps:
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Mark a particular point, say P, and draw a horizontal line on it.
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Using a protractor, make a 30 degree angle with P as the center.
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Draw a line using a ruler to make the angle.
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Make an arc with a compass on the line of the angle at any given length and name this as the Q point.
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From the Q point, draw a perpendicular line to the base and mark the point where it insects the base at R.
The right-angled triangle has been drawn, and now the value of cos 30 can be found out.
Using the Trigonometric Approach
By using the cos square identity in trigonometry i.e., cos2θ = 1 – sin2θ , 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
Conclusion
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 frac{pi}{6}] is [frac{sqrt{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 frac{pi}{6}] is the same in all cases.