Cos 60 Degrees: Value of cos 60 with proof, Examples and FAQ

Cos 60 degrees

In a 60 degrees right-angled triangle, the cosine of angle 60° is a value representing the ratio of the length of the adjacent side (to 60°) to the length of the hypotenuse.

In trigonometry, we write the exact value of cos 60° mathematically. Its exact value in fraction form is ½ equal to 0.5 in the decimal form. Therefore, we write it in the following form in trigonometry.

cos (60°) = cos π/3 = ½ = 0.5

Proof

The exact value of cos 60 degree can be derived using three methods explained below.

• Theoretical Method

To find the value of the cosine of angle 60 degrees, let us consider an equilateral triangle given below:

Since all sides of an equilateral triangle are equal, AB = BC = AC, and AD is perpendicular bisector, dividing BC into two equal parts.

Let us consider the length of each side of the triangle as 2 units.

That is AB = AC = BC = 2 units and CD = BD = 2/2 = 1 unit.

In the △ABC,

The value of cos (60°) = adjacent side/hypotenuse = BD/AB = ½

Similarly, we can determine the value of sin 60° by evaluating the required sides.

In right triangle ABD, Using the Pythagoras theorem:

AB² = AD² + BD²

2² = AD² + 1²

AD² = 2² -1²

AD² = 4 – 1

AD² = 3

AD = √3

Now, 

Sin 60° = opposite side/hypotenuse = AD/AB = √3/2

• Practical Method

You can also find the value of cos of angle 60° practically by constructing a right-angled triangle with 60° angle by geometrical tools.

Draw a straight horizontal line from Point A and then construct an angle of 60° using the protractor.

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Set compass to any length by a ruler. Here, the compass is set to 4.5 cm. Now, draw an arc on the 45° angle line from point A.

Finally, draw a perpendicular line on the horizontal line from point D, and it intersects the horizontal line at point E perpendicularly. Thus, a right-angled triangle ∆ADE is formed. 

Now, calculate the value of the cosine of 60 degrees and for this, measure the length of the adjacent side by a ruler. You will observe that the length of the adjacent side is 2.3 cm. The length of the hypotenuse is taken as 4.5 cm in this example.

Now, find the ratio of lengths of the adjacent side to the hypotenuse and get the value of the cosine of angle 60°.

cos (45°) = AE/AD = (2.3)/(4.5)

So, cos (45°) = 0.5111… ≈ 0.5

• Trigonometric Method

We can prove the value of cos (60°) with a trigonometric approach.

We know that Sin 60° = √3/2

Also, by trigonometric identities,

\sin ^{2} x+\cos ^{2} x=1
Or \cos ^{2} x=1-\sin ^{2} x

Put x = 60°

\cos ^{2} 60^{\circ}=1-\sin ^{2} 60^{\circ}

Put the value of sin 60° 

\cos ^{2} 60^{\circ}=1-(\sqrt{3} / 2)^{2}
\cos ^{2} 60^{\circ}=1-3 / 4

cos (60°) = \sqrt{1/4} = 1/2

Hence, we proved the value of cos (60°) using different approaches.

Example

1. Evaluate: cos 60° + sin 30°

Solution:

We know that cos (60°) = sin (30°) = 1/2.
So, cos (60°) + sin (30°)
= 1/2 + 1/2
= 1

2. Evaluate: 2 sin 60° – 4 cos 60°

Solution:

We know that cos (60°) = ½ and sin (60°) = √3/2
So, 2 sin (60°) – 4 cos (60°)
= 2 (√3/2) – 4(1/2)
= √3 – 2