sin 2A in Terms of A

We will learn to express trigonometric function of sin 2A in terms of A. We know if A is a given angle then 2A is known as multiple angles.


How to proof the formula of sin 2A is equals 2 sin A cos A?

We know that for two real numbers or angles A and B,

sin (A + B) = sin A cos B + cos A sin B

Now, putting B = A on both sides of the above formula we get,

sin (A + A) = sin A cos A + sin A cos A

⇒ sin 2A = 2 sin A cos A

Note: In the above formula we should note that the angle on the R.H.S. is half of the angle on L.H.S. Therefore, sin 60° = 2 sin 30° cos 30°. 

The above formula is also known as double angle formulae for sin 2A.


Now, we will apply the formula of multiple angle of sin 2A in terms of A to solve the below problems.

1. Express sin 8A in terms of sin 4A and cos 4A

Solution:

sin 8A

= sin (2 ∙ 4A)

= 2 sin 4A cos 4A, [Since, we know sin 2A = 2 sin A cos A]


2. If sin A = \(\frac{3}{5}\) find the values of sin 2A.

Solution:

Given, sin A = \(\frac{3}{5}\)

We know that, sin\(^{2}\) A + cos\(^{2}\) A = 1

                                  cos\(^{2}\) A = 1 - sin\(^{2}\) A

                                  cos\(^{2}\) A = 1 - (\(\frac{3}{5}\))\(^{2}\)

                                  cos\(^{2}\) A = 1 - \(\frac{9}{25}\)

                                  cos\(^{2}\) A = \(\frac{25 - 9}{25}\)

                                  cos\(^{2}\) A = \(\frac{16}{25}\)

                                  cos A = √\(\frac{16}{25}\)

                                  cos A = \(\frac{4}{5}\)

   sin 2A

= 2 sin A cos A

= 2 ∙ \(\frac{3}{5}\) ∙ \(\frac{4}{5}\)

= \(\frac{24}{25}\) 


3. Prove that,16 cos \(\frac{2π}{15}\)  cos \(\frac{4π}{15}\)  cos \(\frac{8π}{15}\)   \(\frac{16π}{15}\) = 1.

Solution: 

Let, \(\frac{2π}{15}\) = θ

L.H.S = 16 cos \(\frac{2π}{15}\)  cos \(\frac{4π}{15}\)  cos \(\frac{8π}{15}\)   \(\frac{16π}{15}\) = 1.

= 16  cos θ cos 2θ cos 4θ cos 8θ, [Since, θ = \(\frac{2π}{15}\)]

= \(\frac{8}{sin  θ}\) (2 sin θ cos θ) cos 2θ cos 4θ cos 8θ 

= \(\frac{4}{sin  θ}\) (2 sin 2θ cos 2θ) cos 4θ cos 8θ 

= \(\frac{2}{sin  θ}\) (2 sin 4θ cos 4θ) cos 8θ 

= \(\frac{1}{sin  θ}\) (2 sin 8θ cos 8θ)

= \(\frac{1}{sin  θ}\) ∙ sin 16θ

= \(\frac{1}{sin  θ}\) ∙ sin (15θ + θ)

= \(\frac{1}{sin  θ}\) ∙ sin (2π + θ), [Since, \(\frac{2π}{15}\) = θ 15θ = 2π]

= \(\frac{1}{sin  θ}\) ∙ sin (θ), [Since, sin (2π + θ) = sin θ]

= 1 = R.H.S.                Proved

 Multiple Angles






11 and 12 Grade Math

From sin 2A in Terms of A to HOME PAGE




Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.



New! Comments

Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.




Share this page: What’s this?

Recent Articles

  1. Multiplication Table | Learn Tables from 0 – 25 | Multiplication Table

    Jan 14, 25 11:53 PM

    multiplication table
    In math multiplication table we will learn the tables from 0 – 25. These multiplication tables help the students to learn the essential multiplication facts. Multiplication tables are very important f…

    Read More

  2. 3rd Grade Math Worksheets |3rd Grade Math Sheets|3rd Grade Math Lesson

    Jan 14, 25 11:02 PM

    3rd Grade Math Worksheets
    3rd grade math worksheets is carefully planned and thoughtfully presented on mathematics for the students. Teachers and parents can also follow the worksheets to guide the students.

    Read More

  3. 3rd Grade Subtraction Worksheet | 3-Digit Subtraction Worksheets | Ans

    Jan 14, 25 01:57 PM

    Fill in the Missing Numbers Subtraction and Addition
    In 3th Grade Addition Worksheet we will solve how to subtract 3-digit numbers by expansion, subtraction of 3-digit numbers without regrouping, subtraction of 3-digit numbers with regrouping, propertie…

    Read More

  4. Facts about Subtraction | Subtraction of Small Numbers|Solved Examples

    Jan 14, 25 12:29 AM

    The operation to finding the difference between two numbers is called subtraction. Let us know some facts about subtraction which will help us to learn subtraction of large numbers. 1. Subtraction wit…

    Read More

  5. Word Problems on Subtraction |Worksheet on Subtraction Word Problems |

    Jan 14, 25 12:21 AM

    Subtraction Problem
    In word problems on subtraction we need to read the question carefully and understand what we need to find out. We know, in subtraction the larger number from which we subtract the other number (the s…

    Read More