In trigonometry, the cosine function is defined as the ratio of the adjacent side to the hypotenuse. If the angle of a right triangle is equal to 30 Â degrees, and then the value of cosine at this angle i.e., the value of Cos 30 degree is in a fraction form as âˆš3/2. It is noted that the value ofÂ Sec 30 will be reciprocal of Cos 30 value.
Cos 30 Value
Cos 30 degrees is written as cos 30Â°Â and has a value in fraction form as âˆš3/2.
Cos 30Â°Â = âˆš3/2
Cos 30Â°Â = âˆš3/2 is an irrational number and equals to 0.8660254037 (decimal form).
Therefore, the exact value of cos 30 degrees isÂ written as 0.8660 approx.
âˆš3/2 is the value of Cos 30Â° which is a trigonometric ratio or trigonometric function of a particular angle.
Cos 30
Another alternative form of Cos 30Â°Â is pi/6 or Ï€/6 or Cos 33 (â…“)^{g}
Form | Formula | Value |
Trigonometric ratio | âˆš3 / 2 | 0.8660254037 |
Circular system | pi/6 or Ï€/6 | 0.8660254037 |
Centesimal system | Cos 33 (â…“)^{g} | 0.8660254037 |
Cos 30 Degrees Proof
Now that you know the value of Cos 30 degrees, letâ€™s explore how to derive this value.
We will study the two approaches to derive it.
- Theoretical approach
- Practical approach
Theoretical approach
Knowing the property that length of an opposite side is half of the length of the hypotenuse – Right triangle property for an angle equal to 30 degrees in the given right triangle.
As we know two sides those are the length of the opposite side and hypotenuse but the third side is unknown i.e adjacent side. We need to find this side to find the value of Cos 30 degrees.
We will use the Pythagorean theorem to find the value of this third side
In triangle AOB,
AO ^{2} = AB^{2 }+ AB^{2}
Hypotenuse = d
Opposite side = d/2
d^{2} = (d/2)^{2} + AB^{2}
d^{2} = d^{2}/4 + AB^{2}
d^{2} – d^{2}/4 = AB^{2}
(4d^{2} – d^{2})/4 = AB^{2}
3d^{2}/4 = AB^{2}
Sqrt (3) x d/2 = AB
AB/d = âˆš(3) / 2
As d is the length of the opposite side, so
Length of adjacent / length of hypotenuse = âˆš3 / 2
The value of angle BOA in the given right triangle is pi/6 and the given ratio as per the definition of cosine ratio represents the value of Cos 30 degrees
Cos 30Â°Â =Length of adjacent/length of hypotenuse = âˆš3/2 = 0.8660254037
Practical Approach
Another way to calculate the value of Cos 30 degree is through a practical approach. So if we make a right triangle using constructions, Cos pi/6 can be calculated.
Step 1: Mark a point P on the plane and horizontally draw a line on it.
Step 2: With the help of protractor makes an angle of 30^{0 } with the centre as P and taking baseline as the drawn line in step 1.
Step 3: Use the ruler to draw a line to make an angle of 30 degrees.
Step 4: Using compass draw an arc on the line of angle 30 degrees with any length. Mark that point as Q.
Step 5: From Q draw a line perpendicular to the base (horizontal line). Mark it as R (Intersecting point of the perpendicular line to the base.)
Now after drawing a right-angled triangle POQ, we can calculate the value of Cos 30 degrees.
Cos 30Â°Â =Length of the adjacent/length of the hypotenuse = OP/OQ.
In the above construction, the length of the side OP is taken as 7.5 cm and the length of the adjacent side is unknown. But if we measure it with a ruler, It comes out to be 6.5 cm.
Cos 30Â°Â = OP/OQ = 6.5/7.5 = 0.866666â€¦
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