## Double Angle Formulas

In this page we are going to see double angle formulas and also we are going to see the example problems.

1.Sin 2A = 2 Sin A cos A

2.Cos 2A = cos² A - Sin² A

3. tan 2A = 2 tan A/(1-tan² A)

4.Cos 2A = 1 - 2Sin² A

5.Cos 2A = 2Cos² A - 1

6. sin 2A = 2 tan A/(1+tan² A)

7.cos 2A = (1-tan² A)/(1+tan² A)

8. sin²A = (1-Cos 2A)/2

9.Cos²A = (1+Cos 2A)/2

Now we are going to see the example problems based on the above formulas.

Example 1:

Prove that 2 Sin 15° cos 15° = 1/2

L.H.S

2 Sin 15° cos 15°

This looks like the formula 2 Sin A cos A.Now we have to apply the formula.The required formula is

Sin 2A = 2 Sin A cos A

Instead of A we have 15°

2 Sin 15° cos 15° =  Sin 2(15°)

=  Sin 30°

=  1/2

R.H.S

Hence proved             double angle formulas

Example 2:

Show that Cos 20° cos 40° cos 80° = 1/8

L.H.S

Cos 20° cos 40° cos 80°

=Cos 20° cos (60° - 20°) cos (60° + 20°)

=Cos 20° cos (60° - 20°) cos (60° + 20°)

Cos (A-B) Cos (A+B) = Cos² A - Sin² A

= Cos 20°[cos ² 60°- Sin² 20°]

= Cos 20°[(1/2)² - Sin² 20°]

= Cos 20°[1/4 -Sin² 20° ]

= Cos 20°(1 -4Sin² 20°)/4

= Cos 20°(1 -4(1-cos² 20°)/4)

= Cos 20°(1 -4 + 4cos² 20°)/4

= Cos 20°(-3 + 4cos² 20°)/4

= (-3Cos 20° + 4cos³ 20°)/4

= ( 4cos³ 20° - 3Cos 20°)/4

= Cos (3 x 20°)/4

= Cos (3 x 20°)/4

=  Cos 60°/4

=  (1/2) /4

=  1/8

R.H.S

Hence proved

Related Topics

Quote on Mathematics

“Mathematics, without this we can do nothing in our life. Each and everything around us is math.

Math is not only solving problems and finding solutions and it is also doing many things in our day to day life.  They are:

It divides sorrow and multiplies forgiveness and love.

Some people would not be able accept that the subject Math is easy to understand. That is because; they are unable to realize how the life is complicated. The problems in the subject Math are easier to solve than the problems in our real life. When we people are able to solve all the problems in the complicated life, why can we not solve the simple math problems?

Many people think that the subject math is always complicated and it exists to make things from simple to complicate. But the real existence of the subject math is to make things from complicate to simple.”

Double Angle Formulas to Trigonometry

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