Half Angle Formulas





In this page half angle formulas we are going to see formulas using half angle in trigonometry.

  • Sin A = 2 Sin (A/2) cos (A/2)
  • Cos A = cos² (A/2) - Sin² (A/2)
  •  tan A = 2 tan (A/2)/[1-tan² (A/2)]
  • Cos A = 1 - 2Sin² (A/2)
  • Cos A = 2Cos² (A/2) - 1
  •  sin A = 2 tan (A/2)/[1+tan² (A/2)]
  • cos A = [1-tan²(A/2)]/[1+tan² (A/2)]
  •  sin²A/2 = (1-Cos A)/2
  • Cos²A/2 = (1+Cos A)/2
  •  tan²(A/2) = (1-Cos A)/(1+Cos A)


Now let us see the example problems using the above formulas

Example 1:

Using half angle find the value of Sin 15°

Solution:

We can write 15° as 30°/2

So Sin 15° = Sin 30°/2.This looks like the formula Sin (A/2) and the required formula is

            sin²A/2 = (1-Cos A)/2

            sin (A/2) = [(1-Cos A)/2]


           Sin 30°/2  = √[(1-Cos 30)/2]

                        = √[(1-√3/2)/2]

                        = √[(2-√3)/4]                   

                        = √(2-√3)/2


Example 2:

Using half angle formula find the value of tan 15°

Solution:

We can write 15° as 30°/2

So tan 15° = tan 30°/2.This looks like the formula tan (A/2) and the required formula is

            tan²(A/2) = (1-Cos A)/(1+Cos A)

            tan(A/2)  = (1-Cos A)/(1+Cos A)


           tan 30°/2  = √[(1-Cos 30°)/(1+cos 30°)]

                        = √[(1-√3/2)/(1+√3/2)]

                        = √{ [ (2-√3) / 2 ] / [ (2+√3) / 2 ] }

                        = √[( 2-√3 )/2] x [2/(2+√3)]

                        = √(2-√3)/(2+√3 )                  

                        = √[(2-√3)/(2+√3)] x (2-√3)/(2-√3)                 

                        = √[(2-√3)²/(2² - (√3)²]

                        = √[(2²+(√3)² - 2(2)(√3) /(2² - (√3)²]

                        = √[(2²+(√3)² - 2(2)(√3) /(4-3]

                        = √[(2-√3)² /1]

                        = √[(2-√3)²

                        = 2-√3

Therefore the value of tan 15° is 2-√3


Example 3:

Using half angle find the value of Cos 15°

Solution:

We can write 15° as 30°/2

So Cos 15° = Cos 30°/2.This looks like the formula Cos (A/2) and the required formula is

            Cos²A/2 = (1+Cos A)/2

            Cos (A/2) = √(1+Cos A)/2 


           Cos 30°/2  = √[(1+Cos 30)/2]

                        = √[(1+√3/2)/2]

                        = √[(2+√3)/4]                   

                        = √(2+√3)/2

Related Topics

  1. Trigonometric Ratios
  2. Trigonometric Identities
  3. Complementary Angles In Trigonometry
  4. Values Of Certain Angles
  5. Heights And Distances
  6. Double Angle Formulas
  7. Compound Angle Formulas
  8. 3A formulas
  9. Compound angles sum and differences
  10. Sum to product forms
  11. Trigonometry Problems Using Identities
  12. Trigonometry Practical Problems

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 subtracts sadness and adds happiness in our life.    

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.”








Half Angle Formulas to Trigonometry
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