Cos 60 degrees

What is Trigonometry and What are Trigonometric Ratios?

  • Trigonometric ratios in trigonometry are derived from the three sides of a right- angled triangle basically the hypotenuse, the base(adjacent) and the perpendicular(opposite).

  • According to trigonometric ratio in maths, there are three basic or primary trigonometric ratios also known as trigonometric identities.

  • Trigonometric ratios help us to find missing angles and missing sides of a triangle.

  • To be more specific, they are used in right- angled triangles, the triangles with one angle equal to 90 degrees

Here they are!

The primary trigonometric functions used are cosine, sine and tangent. Cos 60 degree value and other trigonometric ratios are used for common angles like 0°, 30°,45°, 60°,  90° are used in trigonometric equations and calculations.

NAME

ABBREVIATION

RELATIONSHIP

Sine

Sin

Opposite/Hypotenuse

Cosine

Cos

Adjacent/Hypotenuse

Tangent

Tan

Opposite/Adjacent


A Little Information About What a Right - Angled Triangle is?

Consider a right-angle triangle ABC, with its three sides namely the opposite, adjacent and the hypotenuse. In a right-angled triangle we generally refer to the three sides in order to their relation with the angle . The little box in the right corner of the triangle given below denotes the right angle which is equal to 90°. (image will be updated soon)


The 3 Sides of a Right - Angled Triangle are as Follows-

  • The side opposite to the right angle is the longest side of the triangle which is known as the hypotenuse(H). 

  • The side that is opposite to the angle \[\theta \] is known as the opposite(O). 

  • The side which lies next to the angle  \[\theta \] is known as the Adjacent(A)

     According to the Pythagoras theorem, 

In a right-angle triangle,

\[{\left( {Opposite} \right)^2} + {\left( {Adjacent} \right)^2} = {\text{ }}{\left( {Hypotenuse} \right)^2}\]


What Cosine Function is?

Cosine of an angle is defined as the ratio of the side which is adjacent to the angle (θ) to the hypotenuse (longest side) in the triangle.


Cos 60 Degree Value –

In a right - angled triangle, the value of cosine of an angle is defined as the ratio of the length of the adjacent (which is also known as the base) to the angle to the longest side of the triangle which is known as the hypotenuse.

Let us take a right -angled triangle, (image will be updated soon)

Seeing the triangle given above, 

\[\cos \left( \theta  \right) = \frac{{Adjacent}}{{Hypotenuse}}\]

\[\cos \left( \theta  \right) - \frac{b}{h}\]

Now, to find the cos 60 value, we will consider an equilateral triangle ABC (all sides are equal).(image will be updated soon)

The angle marked in blue in the above equilateral is equal to 60 degrees. 

Since the triangle ABC is an equilateral triangle, AB=BC=AC and AD is perpendicular bisecting the side BC into two exactly equal parts.

In the ∆ABC, AB, BC and AC are equal as they are sides of an equilateral triangle. 

Now, we will consider that the sides AB, BC, AC have a measurement of 2 units and let us consider that sides BD and CD measure 1 unit.

According to Pythagoras theorem, 

\[{\left( {Opposite} \right)^2} + {\left( {Adjacent} \right)^2} = {\text{ }}{\left( {Hypotenuse} \right)^2}\]


In a right-angle triangle ABD,

(AB)2= (AD)2    + (BD)2

(2)2= (AD)2    + (1)2

(AD)2 = (2)2 – (1)2

(AD)2 = 4-1 = 3

AD = √3

We have got all the sides of the right -angled triangle ABD.

Now, we can write the cos 60 value easily,

The value of adjacent = 1 

The value of opposite =  \[\sqrt 3 \]

The value of hypotenuse = 2

As we know, \[\cos \left( {{{60}^o}} \right) = \frac{{Adjacent}}{{Hypotenuse}} = \frac{1}{2}\]

Similarly, we can calculate the value of sin and tan,

\[\sin \left( \theta  \right) = \frac{{opposite}}{{Hypotenuse}}\]

\[\sin \left( {{{60}^o}} \right) = \frac{{\sqrt 3 }}{2}\]

\[\operatorname{Tan} \left( \theta  \right) = \frac{{Opposite}}{{Adjacent}}\]

\[\tan \left( {{{60}^o}} \right) = \frac{{\sqrt 3 }}{2}\]

We can calculate the values of sine, cosine and tangent with respect to all the angles.

Here’s a table showing the value of each ratio with respect to different angles. These ratios are basically used in different calculations and are important for solving various problems.

Table showing the different values of sin, cos and tan for different angles.


Angle

0

degrees

30 degrees

45

degrees

60 degrees 

90 degrees

180 degrees

270 degrees

360 degrees

Sin

    0

\[\frac{1}{2}\]

\[\frac{1}{{\sqrt 2 }}\]

\[\frac{{\sqrt 3 }}{2}\]

  1

0

-1

0

Cos

    1

\[\frac{{\sqrt 3 }}{2}\]

\[\frac{1}{{\sqrt 2 }}\]

\[\frac{1}{2}\]

  0

-1

0

1

Tan

    0

\[\frac{{\sqrt 3 }}{2}\]

      1

      \[\sqrt 3 \]

  ∞

0

0

Cot

    ∞

      \[\sqrt 3 \]

      1

    \[\frac{1}{{\sqrt 3 }}\]

  0

0

Cosec

    ∞

         

2

    \[\sqrt 2 \]

      \[\frac{2}{{\sqrt 3 }}\]

 

1

-1

Sec

    1

    \[\frac{2}{{\sqrt 3 }}\]

\[\sqrt 2 \]

      

2

-1

1


Table Showing Different Values of Sin, Cos and Tan with Respect to Radians.

Angle

0 degrees

    30 degrees

    45 degrees

  60 degrees

  90

degrees

  180 degrees

  270 degrees

  360 degrees

Radian 

0

\[\frac{\pi }{6}\]

\[\frac{\pi }{4}\]

\[\frac{\pi }{3}\]

\[\frac{\pi }{2}\]

\[\pi \]

\[\frac{{3\pi }}{2}\]

\[2\pi \]


A few Applications of Trigonometry.

  1. Trigonometry is used in cartography which is the creation of maps.

  2. Trigonometry is used in aviation industries.


Questions on Cos 60 Value –

Question 1) Give a short derivation of cos 60 degree.

Solution) Let us consider a right- angled triangle with one angle as 60°. The other two angles of the triangle are 90°,30° .

For a triangle with angles 60°,90°,30°  the sides are always in the ratio 1: 2: \[\sqrt 3 \] (image will be updated soon)

In a right –angled triangle,

             \[\cos {60^o}\frac{{Adjacent}}{{Hypotenuse}} = \frac{1}{2}\]

Therefore, the value of Cos 60 degrees  =\[\frac{1}{2}\]


Question 2) Calculate cos(A) from the triangle given below. (image will be updated soon)

Solution) We know the formula of \[\cos \left( A \right) = \frac{{Adjacent}}{{Hypotenuse}}\]

In the given question, 

Adjacent = 12

Hypotenuse = 13

Then, \[\cos \left( A \right) = \frac{{12}}{{13}}\]