cos30degrees=square root of 3/2?

Question

I want to know how did they get square root of 3/2....I need a good solution.

Answer 1

draw a right-angled triangle with sides 1 and sqrt(3) [approx. = 1.732], and hypotenuse 2. (pythagoras' theorem: 1² + [sqrt(3)]² = 2²)

now measure the angle between the side with length sqrt(3) and the hypotenuse, it will definitely read 30deg (± a few degrees if your drawing is inaccurate).

since cos (theta) = adjacent / hypotenuse
cos 30deg = sqrt(3) / 2

extra info: with this diagram, you can also show that:

sin 30deg = cos 60deg = 1/2 = 0.500
sin 60deg = cos 30deg = sqrt(3) / 2 = 0.866
tan 30deg = 1 / sqrt(3) = 0.577
tan 60deg = sqrt(3) = 1.732

try it yourself, and in case you've forgotten, here's a little something:

sin(theta) = opposite / hypotenuse
cos(theta) = adjacent / hypotenuse
tan(theta) = opposite / adjacent

Answer 2

Consider an equilateral triangle. Its sides are equal and its angles are all 60°. Now bisect any angle, and you will have bisected the side opposite the angle, forming a 30°, 60°, 90° triangle. The hypoteneuse will be the original side length, the side opposite the 30° angle will be 1/2 the original side length. By the theorem of Pythagorus, the side adjacent to the 30° angle will be
x = (s^2 - (s/2)^2)^1/2
x = (s^2 - s^2/4)^1/2
x = s(1 - 1/4)^1/2
x = s((4 - 1)/4)^1/2
x = (1/2)s(3)^1/2

cos(30°) = x/s = (1/2)(s/s)(3)^1/2 = (1/2)√3
cos(30°) = (√3)/2

Answer 3

There is a proof; Draw a right angled triangles with one angle 30 degrees. The other angle is therefore 60 degrees. Then construct a similar triangle adjacent to this one with one side of angle 30 touching. These will be similar triangles. Using properties of similar triangles and the Pythagoras theorem you can get expression for all trigonometric ratios of angles 30 & 60.

Answer 4

It turns out that a right-angled triangle with a 30 degree corner and a hypotenuse of 2 has side adjacent to the 30 degrees of sqrt3.

Answer 5

u need to draw a right angled triangle to see it.

there will be one adjacent,one opposite line, and one hypothenus. then use pythgoras rule. a^2 +b^2 = c^2

cos = adjacent/hypotenus.

so whatever value u put in as long as your degree (ie. 30) is in the angle the cosine value will turn out the same.

Answer 6

I would use the fact that 60 degrees are twice 30 degrees:

cos(30) = sin(60) = 2sin(30)cos(30) // divide by cos(30)

1 = 2sin(30)

sin(30) = 0.5

As you know (cos x)^2 + (sin x)^2 = 1

x=30 and sin(x)=0.5, hence

cos(30)^2 + 0.5^2 = 1

cos(30)^2 = 1-0.5^2

cos(30)^2 = 0.75

Now, because between 0 and 90 degrees cos is non-negative:

cos(30) = sqrt(0.75) = sqrt(3/4) = sqrt(3)/sqrt(4) = sqrt(3)/2