Cos 30 Degrees: Value of cos 30 with proof, Examples and FAQ

Cos 30 degrees

The value of cosine if the angle of the right-angled triangle equals 30 degrees is called cos of angle 30 degrees. The cosine of angle 30° is a value representing the ratio of the length of the adjacent side (of the considered angle) to the length of the hypotenuse.

In trigonometry, we write cos 30° mathematically, and its exact value in fraction form is √3/2. Therefore, we write it in the following form in trigonometry.

cos (30°) = cos π/6 = √3/2

Value of Cos 30°

The exact cosine value of angle 30 degrees is √3/2 equal to 0.8660254037… in decimal form. The approximate value of the cosine of angle 30 is equal to 0.8660.

Cos (30°) = 0.8660254037… ≈ 0.8660

Proof

The exact value of cos π/6 can be derived using three methods explained below.

• Theoretical Method

We must know the relation between the sides of a right triangle when one of its angles is 30°. According to this property, the length of the opposite side is half of the hypotenuse length (to angle 30°,i.e, the perpendicular).

We will evaluate the exact value of the cosine of angle 30 degrees by using this property.

In this case, we are considering the length of the hypotenuse as ‘d’, then the length of the opposite side (to angle 30°,i.e, the perpendicular), PQ will be ‘d/2’.

Now, the lengths of the opposite side and hypotenuse are known, but the length of the adjacent side (to angle 30°), OQ is unknown.

It is essential to find the value of cos (30°). 

So, apply the Pythagorean Theorem to find the value of the adjacent side.

\text { Hypotenuse }^{2}=\text { Perpendicular }^{2}+\text { Adjacent Side }^{2}

\mathrm{OP}^{2}=\mathrm{OQ}^{2}+\mathrm{PQ}^{2}
\Rightarrow \mathrm{d}^{2}=\mathrm{OQ}^{2}+(\mathrm{d} / 2)^{2}
\Rightarrow \mathrm{d}^{2}=\mathrm{OQ}^{2}+\mathrm{d}^{2} / 4
\Rightarrow \mathrm{OQ}^{2}=\mathrm{d}^{2}-\mathrm{d}^{2} / 4
\Rightarrow \mathrm{OQ}^{2}=(3 / 4) \mathrm{d}^{2}
\Rightarrow \mathrm{OQ}=(\sqrt{3} / 2) \mathrm{d}
\Rightarrow \mathrm{OQ} / \mathrm{d}=\sqrt{3} / 2

Here d represents the length of hypotenuse.

⇒ Length of adjacent side/Hypotenuse = √3/2 

Therefore, we can write that cos (30°) = √3/2

• Practical Method

We can find the value of the cosine of angle 30° by constructing a right-angled triangle with a 30° angle by using geometrical tools.

Draw a straight horizontal line from Point G and then construct an angle of 30° using the protractor.

Set compass to any length by a ruler. Here, the compass is set to 7.5 cm. Now, draw an arc on the 30° angle line from point G, and it intersects the line at point H.

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

Now,  calculate the value of the cosine of 30 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 6.5 cm. The length of the hypotenuse is taken as 7.5 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 30°.

cos (30°) = GI/GH = 6.5/7.5

So, cos (30°) = 0.866666… ≈ 0.8660

• Trigonometric approach

We can prove the value of cos (30°) with a trigonometric approach.

we know that, sin 30° = 1/2

Also, by trigonometric identities,

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

Put x = 30°

\cos ^{2} 30^{\circ}=1-\sin ^{2} 30^{\circ}

Put the value of sin 30° 

\cos ^{2} 30^{\circ}=1-(1 / 2)^{2}

\cos ^{2} 30^{\circ}=3 / 4
\cos \left(30^{\circ}\right)=\sqrt{(3 / 4)}=\sqrt{3} / 2

 Hence, we proved the value of cos (30°) using different approaches.

Example

1. Evaluate: cos 30° + sin 60°

Solution:

We know that cos (30°) = sin (60°) = √3/2
So, cos (30°) + sin (60°)
= √3/2 + √3/2
= 2(√3/2)
= √3

2. Evaluate: 2 cos 30° – 2 sin 30°

Solution:

We know that cos (30°) = √3/2,
and sin (30°) = 1/2
So, 2 cos (30°) – 2 sin (30°)
= 2 (√3/2) – 2(1/2)
= √3 – 1