Prove the double angle formula through the following method

In summary, the conversation is about proving the equation sin^2(a/2) = (1-cosa)/(2) using the double angle formula for cosine. The first step involves understanding that 2*(a/2) = a and then using the formula for cos(2beta) with beta = (a/2). The second step involves using the formula cos^2(b) + sin^2(b) = 1 to express cos(2beta) only in terms of sin^2(b). Finally, the last step is to replace b with a/2 to complete the proof.
  • #1
needingtoknow
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Homework Statement



I have to prove that sin^2(a/2) = (1-cosa)/(2)

cosa = cos(2*(a/2)) = 1 -2sin^2(a/2)

I don't understand this step that was given in the solutions how do I get it?
 
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  • #2
Assuming it is clear to you that ## 2*\frac{\alpha}{2}=\alpha##, the equality is obtained using the formula for ##\cos(2\beta)## with ##\beta=\frac{\alpha}{2}##
 
  • #3
Yes I understand that they have used the double angle formulas to derive hence cos(2*(a/2)). So the formulas for it is then cos2a = cos^2(a) - sin^2(a), so does that mean I have to do [cos(a/2)]^2 - [sin(a/2)]^2 or what then? That is what is confusing me.
 
  • #4
First of all, don't use all the same letter, otherwise you will be confused about who is who in like a second... this is always good rule in math...
Now assuming b=a/2, you have cos(2b)=cos^2(b)-sin^2(b) indeed... now you can use the fundamental formula cos^2(b)+sin^2(b)=1 and express cos(2b) only as function of the sin^2(b). Done this, all is straightforward replacing b with a/2
 

1. What is the double angle formula?

The double angle formula is a mathematical identity that expresses the relationship between the sine, cosine, and tangent of a double angle in terms of the sine, cosine, and tangent of the original angle. It is commonly used in trigonometry and calculus.

2. What is the method used to prove the double angle formula?

The method used to prove the double angle formula involves manipulating the equations of the sine, cosine, and tangent of a double angle using basic trigonometric identities and algebraic properties.

3. Why is it important to prove the double angle formula?

Proving the double angle formula is important because it provides a deeper understanding of the relationships between trigonometric functions and allows for the derivation of other useful identities. It is also essential for solving more complex trigonometric equations and problems.

4. How can the double angle formula be applied in real-world situations?

The double angle formula can be applied in real-world situations such as in physics and engineering, where it is used to calculate the magnitude and direction of forces, the trajectory of projectiles, and the movement of objects in circular motion.

5. Are there any alternative methods to prove the double angle formula?

Yes, there are alternative methods to prove the double angle formula, such as using geometric proofs or using complex numbers. However, the most commonly used method is the one involving trigonometric identities and algebraic manipulation.

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