Value of Cos 30 Degrees: Formula and Derivation

The value of cos 30 degrees in fraction form is √3/2, and 0.8660254037 in decimal form.

• The cosine function is a periodic function used in trigonometry.
• Trigonometry is a branch of mathematics that deals with angles, their properties, and applications.
• In real-life problems, it is a function that connects the angles and sides of a right-angled triangle.
• It is used to figure out the value of an unknown angle or side.
• When determining the value of the unknown, many trigonometry formulas and identities are used.
• The sine, cosine, tangent, cosecant, secant, and cotangent are all trigonometric functions.

Key Terms: Cos 30 degrees, Radians, Trigonometric Ratios, Trigonometric Table, Trigonometric functions, Cosine functions, Cosine rule, Triangle, Right angle triangle

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Trigonometric Ratios

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The ratios of the sides of a triangle, taken two at a time, are known as trigonometry ratios.

• These proportions show how one side of the triangle relates to the other.
• One of the triangle's angles must be a right angle for trigonometric ratios to be possible.
• The horizontal part of the right angle, or the x-axis, is known as the base of the triangle.
• The vertical hand, or the y-axis, is known as the adjacent side if the angle of reference is horizontally opposite to the right angle.
• The hypotenuse is the segment that connects the ends of both of these hands.

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What is a Cosine Function?

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The cosine function is a periodic function that is used widely in trigonometry.

• The cosine function is the ratio of the lengths of the side adjacent to the angle and the hypotenuse of a right-angled triangle.
• It is defined for acute angles in the context of a right-angled triangle.
• Its value varies as the angle changes,
• The cosine is used for modeling several real-world phenomena, including radio waves, tides, sound waves, musical tones, and electrical currents.
• The cosine function is simply denoted as cos x, where x is the angle.

The unit circle is the easiest method to understand the cosine function.

• Draw a unit circle on the coordinate plane with the angle centered at the origin, with one side as the positive x-axis, for a given angle measure θ.
• The x-coordinate of the point where the other side of the angle intersects the circle is cos(θ), and the y-coordinate is sin(θ).

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Cos 30 degrees Value

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The value of cos 30 degrees in fraction form is √3/2.

\(Cos\:30^{\circ} = \frac{\sqrt 3}{2}\)

In decimal form, the value of cos 30 degrees is an irrational number and equals 0.8660254037, taken as 0.866 approximately.

\(Cos\:30^{\circ} = 0.8660\)

Cos 30° can also be represented as pi/6 or π/6 or Cos 33(1/3)^g in alternative forms.

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Cos 30 Degrees Derivation

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The value of cos 30 degrees can be proven by using two approaches

• Theoretical approach
• Practical approach

Theoretical approach

Knowing the right triangle property the length of the opposite side is half the length of the hypotenuse at an angle of 30 degrees in the provided right triangle.

• We know two sides, the length of the opposite side and the hypotenuse, but the third side, i.e. the adjacent side, is unknown.
• We need to determine this side to determine the value of Cos 30 degrees.

The Pythagoras theorem can be used to get the value of this third side

From triangle AOB, we have

AO² = AB² + OB²

Also, we have

• AO = d
• AB = d/2

On substituting the values, we get

d² = (d/2)² + OB²

⇒ OB = √(3d² / 4)

⇒ OB = (√3/2)d

Hence, the length of the adjacent side, OB = (√3/2)d

Also, the length of the hypotenuse, OA = d

Therefore,

Length of the adjacent side / Length of the hypotenuse = √3/2

The value of angle BOA in the provided right triangle is pi/6, and the given ratio indicates the value of Cos 30 degrees according to the definition of cosine ratio.

Hence, 

Cos 30° = Length of adjacent side / Length of hypotenuse = √3/2 = 0.8660254037

Practical approach

A practical approach can be used to determine the value of Cos 30 degrees.

The below steps should be followed to determine the value of Cos pi/6

• Step 1: Make a point P on the plane and draw a horizontal line through it.
• Step 2: Make an angle of 30° with a protractor, using the center as P and the baseline as the drawn line in step 1.
• Step 3: Draw a line with the ruler to make a 30° angle.
• Step 4: Draw an arc on the line of a 30° angle with any length using a compass. Mark the position with the letter Q.
• Step 5: Draw a line perpendicular to the base (horizontal line) from Q. Mark it as R (Intersecting point of the perpendicular line to the base.)

We can now determine Cos 30 degrees by drawing a right-angled triangle PRQ.

Cos 30° = Length of adjacent side / Length of hypotenuse = PR/PQ

The length of the side PQ in the above figure is 7.5 cm, while the length of the adjacent side is unknown. However, when measured with a ruler, it measures 6.5 cm.

Therefore, 

Cos 30° = PR/PQ = 6.5/7.5 = 0.866666…

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Trigonometry Table

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Things To Remember

• The radius of a unit circle is always taken as one.
• The trigonometric function and values are supposed to deal with relation with the right-angled triangle.
• The value of Cos 30 degrees is √3/2.
• Trigonometry is used to find an angle or a side whose value is not known.
• According to what Pythagoras Theorem states: (Hypotenuse)² = (Base)² + (Altitude)²
• Aviation, Criminology, Marine Biology, And Navigation use Trigonometry.