Cos 60 Degrees In Fraction

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

Cos lx 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 grade is ½ equal to 0.5 in the decimal form. Therefore, we write information technology in the following form in trigonometry.

cos (sixty°) = cos π/iii = ½ = 0.5

Proof

The verbal value of cos 60 degree can be derived using iii methods explained below.

Theoretical Method

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

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

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

That is AB = Air-conditioning = BC = 2 units and CD = BD = ii/2 = 1 unit of measurement.

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:

AB ² = Advertizement² + BD ²

two ² = AD ² + i ²

Ad ² = 2 ² -one ²

Advertizing ² = 4 – i

AD ² = 3

Advertizement = √3

Now,

Sin 60° = opposite side/hypotenuse = Advertisement/AB = √three/two

Practical Method

Yous tin also find the value of cos of angle 60° practically past constructing a right-angled triangle with 60° bending by geometrical tools.

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

Set compass to whatever length by a ruler. Here, the compass is set to 4.five cm. Now, draw an arc on the 45° angle line from bespeak A.

Finally, depict a perpendicular line on the horizontal line from bespeak D, and information technology intersects the horizontal line at signal Due east perpendicularly. Thus, a right-angled triangle ∆ADE is formed.

Now, calculate the value of the cosine of lx degrees and for this, mensurate the length of the adjacent side by a ruler. You will discover that the length of the next side is 2.3 cm. The length of the hypotenuse is taken as iv.5 cm in this instance.

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

cos (45°) = AE/AD = (2.3)/(iv.5)

And so, cos (45°) = 0.5111… ≈ 0.5

Trigonometric Method

We can evidence the value of cos (lx°) with a trigonometric approach.

We know that Sin threescore° = √3/2

Too, by trigonometric identities,

$\sin ^{2} 10+\cos ^{2} x=1$
Or $\cos ^{2} x=1-\sin ^{ii} x$

Put x = 60°

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

Put the value of sin lx°

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

cos (60°) = $\sqrt{i/4}$ = 1/2

Hence, nosotros proved the value of cos (sixty°) using unlike approaches.

Case

ane. Evaluate: cos 60° + sin 30°

Solution:

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

2. Evaluate: ii sin threescore° – four cos 60°

Solution:

We know that cos (60°) = ½ and sin (sixty°) = √3/2
And then, 2 sin (60°) – four cos (threescore°)
= 2 (√3/two) – four(i/2)
= √3 – 2