# Finding sin²22.5 without a calculator

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KingRich

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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?

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?

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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#2

(Original post by

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?

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?

**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?

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?

Notice that sin(22.5) = sin(45/2). How can you rewrite 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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#3

**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?

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?

Note that cos45 = cos(22.5 + 22.5)

and cos(2x)=cos

^{2}x - sin

^{2}x and a few other things.....

Last edited by gdunne42; 1 month ago

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KingRich

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#4

(Original post by

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 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).

**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 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).

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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#5

(Original post by

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)=cos

**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)=cos

^{2}x - sin^{2}x and a few other things.....although I could be wrong

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gdunne42

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#6

(Original post by

Mmm, oh. I think I’m kinda on the right lines with this idea.

although I could be wrong

**KingRich**)Mmm, oh. I think I’m kinda on the right lines with this idea.

although I could be wrong

^{2}x

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.

Last edited by gdunne42; 1 month ago

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#7

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Looks good to me, and similarly with cos2x=1-2sin

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.

**gdunne42**)Looks good to me, and similarly with cos2x=1-2sin

^{2}xPS 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.

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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#8

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#8

(Original post by

Although, surprisingly I found this question straightforward. I’m still confused with sin²22.5.

I have considered that

**KingRich**)Although, surprisingly I found this question straightforward. I’m still confused with sin²22.5.

I have considered that

**sin45=√2/2**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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#9

(Original post by

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

**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

cos(2x)=1-2sin

^{2}x

is a better approach than sin(2x)=2sin(x)cos(x)

for sin

^{2}(22.5)

Last edited by gdunne42; 1 month ago

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#10

(Original post by

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.

**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.

Okay. I believe I have finally found my way!!!

It seems to make more sense after a thousand attempts haha

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#11

(Original post by

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!!!

It seems to make more sense after a thousand attempts haha

**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!!!

It seems to make more sense after a thousand attempts haha

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#12

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Those 1000 attempts will hopefully have taught you a lot......that sticks

**gdunne42**)Those 1000 attempts will hopefully have taught you a lot......that sticks

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