Half-angle Identity

ukumure

ukumure

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The exact value of sin 15 degrees using the half-angle identity.

This is my solution:
2.png

The given answer on the book:
3.png

My solution is different from the given answer.
Can anyone enlighten me on what mistake I put in my solution? Are they just equal or not? Thank you so much!
 
Last edited:
Well, you can always check with a calculator. I don't see a clear direct way of proving they are equal, but you can suppose they aren't equal, multiply both sides by 4, square them... And reach a contradiction
 
ukumure said:
The exact value of sin 15 degrees using the half-angle identity.

This is my solution:
View attachment 24808

The given answer on the book:
View attachment 24805

My solution is different from the given answer.
Can anyone enlighten me on what mistake I put in my solution? Are they just equal or not? Thank you so much!
Click to expand...
For me, your answer and the given answer are both identity and both have a decimal of 0.25881904... I guess, whichever answer you'd choose, it's the same...
 
ukumure said:
The exact value of sin 15 degrees using the half-angle identity.

This is my solution:
View attachment 24808

The given answer on the book:
View attachment 24805

My solution is different from the given answer.
Can anyone enlighten me on what mistake I put in my solution? Are they just equal or not? Thank you so much!
Click to expand...
First, you need to choose one sign for your answer; it will be positive, since the angle is in the first quadrant.

In addition to using a calculator as a quick check to see if the answers agree, you can square both expressions, and see that both are equal to \(\frac{2-\sqrt{3}}{4}\). Since both your (refined) answer and theirs are positive, they are equal.

What I can't figure out is why they gave a form that would not be obtained by the method they required. What they show is what you can get by an angle difference formula. I suspect they cheated.
 
Dr.Peterson said:
First, you need to choose one sign for your answer; it will be positive, since the angle is in the first quadrant.

In addition to using a calculator as a quick check to see if the answers agree, you can square both expressions, and see that both are equal to \(\frac{2-\sqrt{3}}{4}\). Since both your (refined) answer and theirs are positive, they are equal.

What I can't figure out is why they gave a form that would not be obtained by the method they required. What they show is what you can get by an angle difference formula. I suspect they cheated.
Click to expand...
Thank you so much! This help me a lot :)
 
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