Finding sin²22.5 without a calculator

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?

(edited 1 year ago)

Reply 1

Notnek

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?

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).

Reply 2

gdunne42

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?

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.....

(edited 1 year ago)

Reply 3

KingRich

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

Reply 4

KingRich

Original post by gdunne42

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

although I could be wrong

Reply 5

gdunne42

Original post by KingRich

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

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.

(edited 1 year ago)

Reply 6

KingRich

Original post by gdunne42

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

Reply 7

mqb2766

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.

Reply 8

gdunne42

Original post by KingRich

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)

(edited 1 year ago)

Reply 9

KingRich

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!!!

It seems to make more sense after a thousand attempts haha

Reply 10

gdunne42

Original post by KingRich

It seems to make more sense after a thousand attempts haha

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

Reply 11

KingRich

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!