Value of Cos 30: Step-by-Step Guide

What is the Importance and Relevance of cos 30?

What are Trigonometric Ratios?

Trigonometry ratios are the ratios of the sides of a triangle taken two at a time. These ratios give the relationship of one side of the triangle to another. For trigonometric ratios to be feasible one of the angles for the triangle must be a right angle. If we assume that the angle of reference is horizontally opposite to the right angle than the horizontal section or the x-axis of the right angle is termed as the base of the triangle, and the vertical hand or the y-axis is known as the adjacent side. The segment joining the ends of both these hands is termed as the hypotenuse. 

The ratio of the adjacent side w.r.t the hypotenuse is called sine (commonly -sin)

The ratio of the base w.r.t the hypotenuse is called the cosine (commonly-cos)

The ratio of the adjacent side w.r.t the base is termed as tangent (commonly tan) 

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Representing Cos 30°:

When the ratio of the base of the reference angle with the hypotenuse is taken, it gives us the cosine value of the ratio at that specific angle. When this angle is of 30°, then the value is called cos 30. The value of cos 30 is equal to the value of sin 60 since they complement each other within a triangle. Hence, the value of sin 30 can be used with cos 60 too, considering this pattern, we can conclude that sin (90-x) = cos x; where x can be any angle.

Derivative Cos 30°: 

For deriving the value of cos 30°, let us take a look at the triangle formed and analyse it.

If we look at the angles, then, one of the angles is always 90°, and the other two angles remain complementary to each other. If we have to take the value of cos 30° means that one of the angles is 30° and hence the other angle is 180° - (90° + 30°) which is equal to 60°. Now using the property of right angles triangle with one angle 30°, we can see that the hypotenuse is double the length of the adjacent side is half that of the hypotenuse.

Let the hypotenuse be equal to x; then the adjacent side has a value of x/2. 

Using Pythagoras theorem hypotenusee² = base² +adjacent² x²  = adj²+(x/2)² 

Which is equal to,

x² - (x/2)² =adj²

Simplifying this we get,

(¾)x² =adj²

Hence the value of adj²/x² = ¾ ;

Which gives us adj/x = \[\sqrt{3}\]/2

 And since we know the ratio of adjacent with hypotenuse is cosine, hence, the value of cos 30° is \[\sqrt{3}\]/2

Fun Facts: 

Another way you can memorise the value of cos 30° is by remembering its fractional value of π/6 , as denoted in the circular system or the coordinate system.

Sample Problems:

• Problem 1:  Find the other sides of a triangle if the hypotenuse is of 12 units and one angle is 30°

Solution: We know that the value of cos 30° is  \[\sqrt{3}\]/2; Therefore we can use this to calculate the adjacent side of the triangle, i.e.

30° = \[\sqrt{3}\]/2 = adjacent/hypotenuse = x/12

            This implies, x = 12x\[\sqrt{3}\]/2 = 6\[\sqrt{3}\]

            Hence one side is 6\[\sqrt{3}\],

Now, using Pythagoras theorem,hypotenuse = adjacent²+base² 

12² = (6\[\sqrt{3}\])²+ base² →12²-(6\[\sqrt{3}\])² = base²

            base = \[\sqrt{144-108}\] = \[\sqrt{36}\] = 6 ;

Hence the two sides are found out to be 6  and 6 \[\sqrt{3}\].

Problem 2: A Jet plane during testing flies at the speed of 1.5 Mach as soon as it takes off. Meanwhile, it covers a distance of 2000 meters after it was beside you. Now, you have to look at a 30° angle to see the plane and calculate the altitude of the plane.

Solution: Since you are looking at it with an elevation of 30° means that the angle of elevation w.r.t to the ground is 30°. Since the altitude acts as an adjacent side for a right triangle, we can calculate the elevation by using trigonometry.

             Cos Cos 30° = \[\frac{adjacent}{hypotenuse}\] = \[\frac{altitude}{2000}\]

             \[\sqrt{3}\]/2 = \[\frac{altitude}{2000}\] ⇒altitude 1000\[\sqrt{3}\]

Hence we can see that the plane has reached an altitude of 1000\[\sqrt{3}\]m

• Problem 3:  The fire-fighter ladder extended to 20m long makes an angle of 30° with the ground while leaning against a wall. Find out how much a fireman has to walk to reach the ladder steps if he was initially standing at the door on the wall.

Solution: To calculate the distance of the stair from the wall, we need to first find a right triangle with one of the sides as the path of the fireman. The path forms the adjacent of the ladder whereas the ladder itself is the hypotenuse and the wall all along with acts as the base. Hence, we can now apply trigonometry to calculate the adjacent. 

Using cos 30° =   \[\sqrt{3}\]/2

            \[\sqrt{3}\]/2 = \[\frac{adjacent}{hypotenuse}\] = \[\frac{adjacent}{20}\]

             adjacent = 20\[\sqrt{3}\]/2 = 10\[\sqrt{3}\]

Hence the fireman has to walk a distance of 10\[\sqrt{3}\]m to reach the ladder step.