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 = 35 find the values of sin 2A.

Solution:

Given, sin A = 35

We know that, sin2 A + cos2 A = 1

                                  cos2 A = 1 - sin2 A

                                  cos2 A = 1 - (35)2

                                  cos2 A = 1 - 925

                                  cos2 A = 25925

                                  cos2 A = 1625

                                  cos A = √1625

                                  cos A = 45

   sin 2A

= 2 sin A cos A

= 2 ∙ 3545

= 2425 


3. Prove that,16 cos 2π15  cos 4π15  cos 8π15   16π15 = 1.

Solution: 

Let, 2π15 = θ

L.H.S = 16 cos 2π15  cos 4π15  cos 8π15   16π15 = 1.

= 16  cos θ cos 2θ cos 4θ cos 8θ, [Since, θ = 2π15]

= 8sinθ (2 sin θ cos θ) cos 2θ cos 4θ cos 8θ 

= 4sinθ (2 sin 2θ cos 2θ) cos 4θ cos 8θ 

= 2sinθ (2 sin 4θ cos 4θ) cos 8θ 

= 1sinθ (2 sin 8θ cos 8θ)

= 1sinθ ∙ sin 16θ

= 1sinθ ∙ sin (15θ + θ)

= 1sinθ ∙ sin (2π + θ), [Since, 2π15 = θ 15θ = 2π]

= 1sinθ ∙ 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. Worksheet on Area of a Square and Rectangle | Area of Squares & Rectan

    Jul 19, 25 05:00 AM

    Area and Perimeter of Square and Rectangle
    We will practice the questions given in the worksheet on area of a square and rectangle. We know the amount of surface that a plane figure covers is called its area. 1. Find the area of the square len…

    Read More

  2. Area of Rectangle Square and Triangle | Formulas| Area of Plane Shapes

    Jul 18, 25 10:38 AM

    Area of a Square of Side 1 cm
    Area of a closed plane figure is the amount of surface enclosed within its boundary. Look at the given figures. The shaded region of each figure denotes its area. The standard unit, generally used for…

    Read More

  3. What is Area in Maths? | Units to find Area | Conversion Table of Area

    Jul 17, 25 01:06 AM

    Concept of Area
    The amount of surface that a plane figure covers is called its area. It’s unit is square centimeters or square meters etc. A rectangle, a square, a triangle and a circle are all examples of closed pla…

    Read More

  4. Worksheet on Perimeter | Perimeter of Squares and Rectangle | Answers

    Jul 17, 25 12:40 AM

    Most and Least Perimeter
    Practice the questions given in the worksheet on perimeter. The questions are based on finding the perimeter of the triangle, perimeter of the square, perimeter of rectangle and word problems. I. Find…

    Read More

  5. Formation of Square and Rectangle | Construction of Square & Rectangle

    Jul 16, 25 11:46 PM

    Construction of a Square
    In formation of square and rectangle we will learn how to construct square and rectangle. Construction of a Square: We follow the method given below. Step I: We draw a line segment AB of the required…

    Read More