Finding sin²22.5 without a calculator

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KingRich
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#1
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#1
With these specific questions, am I meant to remember the sin values for the so called special angles? Or, is it possible to solve these by other means?

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I tried to look online but people were using half angle identities but I can’t see that within my book. Even if I use the fact that sin 45=√2/2 I’m still unsure how I can apply that. It doesn’t teach me in the section. Is this GCSE stuff?
Last edited by KingRich; 1 month ago
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Notnek
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(Original post by KingRich)
With these specific questions, am I meant to remember the sin values for the so called special angles? Or, is it possible to solve these by other means?

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Views: 17
Size:  27.5 KB

I tried to look online but people were using half angle identities but I can’t see that within my book. Even if I use the fact that sin 45=√2/2 I’m still unsure how I can apply that. It doesn’t teach me in the section. Is this GCSE stuff?
You're right that you need to use the sin(45) exact value.

Notice that sin(22.5) = sin(45/2). How can you rewrite \sin^2(\frac{x}{2}) using an identity?

When a trig question says "without a calculator" they expect you to use the GCSE trig exact values e.g. sin(45).
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gdunne42
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(Original post by KingRich)
With these specific questions, am I meant to remember the sin values for the so called special angles? Or, is it possible to solve these by other means?

Name:  A1AA7E34-BBB2-4CAC-868F-26D7C1BA8BA9.jpeg
Views: 17
Size:  27.5 KB

I tried to look online but people were using half angle identities but I can’t see that within my book. Even if I use the fact that sin 45=√2/2 I’m still unsure how I can apply that. It doesn’t teach me in the section. Is this GCSE stuff?
This appears to be an exercise in using the addition formulae (in the formula book) and applying them to double angle situations.
Note that cos45 = cos(22.5 + 22.5)
and cos(2x)=cos2x - sin2x and a few other things.....
Last edited by gdunne42; 1 month ago
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KingRich
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(Original post by Notnek)
You're right that you need to use the sin(45) exact value.

Notice that sin(22.5) = sin(45/2). How can you rewrite \sin^2(\frac{x}{2}) using an identity?

When a trig question says "without a calculator" they expect you to use the GCSE trig exact values e.g. sin(45).
That’s the problem then. I need to learn the exact values as I don’t recall much from gcse.

sin(45/2)

I know sin (2A) =2sinacosa

if for example sin a/2=sinacosa/2

mmm, so maybe sin (45)cos(45)/2

so, (√2/2)(√2/2) ÷2?? Nope. Wrong

Or should I use this sin(a+b)=sinacosb+cosasinb
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KingRich
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(Original post by gdunne42)
This appears to be an exercise in using the addition formulae (in the formula book) and applying them to double angle situations.
Note that cos45 = cos(22.5 + 22.5)
and cos(2x)=cos2x - sin2x and a few other things.....
Mmm, oh. I think I’m kinda on the right lines with this idea.

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although I could be wrong
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gdunne42
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(Original post by KingRich)
Mmm, oh. I think I’m kinda on the right lines with this idea.

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although I could be wrong
Looks good to me, and similarly with cos2x=1-2sin2x

PS you can always look up the exact values on your calculator if a question along these lines appears in an exam paper and you don't remember them. You just have to recall that exact values exist for some angles.
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KingRich
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(Original post by gdunne42)
Looks good to me, and similarly with cos2x=1-2sin2x

PS you can always look up the exact values on your calculator if a question along these lines appears in an exam paper and you don't remember them. You just have to recall that exact values exist for some angles.
Although, surprisingly I found this question straightforward. I’m still confused with sin²22.5.

I have considered that sin45=√2/2

so, sin²45=1/2

my confusion comes in when I try to use 45/2.

Should I be thinking along the lines of sin 45 = sin 2A, where a=22.5

so, sin 2(22.5)=2sin 22.5 cos 22.5.. ahh lol
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mqb2766
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(Original post by KingRich)
Although, surprisingly I found this question straightforward. I’m still confused with sin²22.5.
I have considered that sin45=√2/2
Thats the "wrong" starting point. Start with
cos(45)
as you can easily map to cos^2(22.5) or sin^2(22.5) depending on which identity you use.
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gdunne42
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(Original post by KingRich)
Although, surprisingly I found this question straightforward. I’m still confused with sin²22.5.

I have considered that sin45=√2/2

so, sin²45=1/2

my confusion comes in when I try to use 45/2.

Should I be thinking along the lines of sin 45 = sin 2A, where a=22.5

so, sin 2(22.5)=2sin 22.5 cos 22.5.. ahh lol
as I'd mentioned above and mqb2766 confirms
cos(2x)=1-2sin2x
is a better approach than sin(2x)=2sin(x)cos(x)
for sin2(22.5)
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KingRich
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(Original post by mqb2766)
Thats the "wrong" starting point. Start with
cos(45)
as you can easily map to cos^2(22.5) or sin^2(22.5) depending on which identity you use.
Right, so in questions like this, I should consider the identity that allows to go In between the identities because starting with sin 2A introduces cos but it becomes somewhat of a dead end.

Okay. I believe I have finally found my way!!!
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It seems to make more sense after a thousand attempts haha
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gdunne42
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(Original post by KingRich)
Right, so in questions like this, I should consider the identity that allows to go In between the identities because starting with sin 2A introduces cos but it becomes somewhat of a dead end.

Okay. I believe I have finally found my way!!!
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It seems to make more sense after a thousand attempts haha
Those 1000 attempts will hopefully have taught you a lot......that sticks :yy:
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KingRich
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(Original post by gdunne42)
Those 1000 attempts will hopefully have taught you a lot......that sticks :yy:
I completely agree! It might not always be clear right away but I know with maths there’s always a solution, so with persistence and failures, there will always be lessons learned along the way!
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