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### Subtracting and Multiplying Sines of Angle Multiples

```Date: 04/01/2010 at 10:30:17
From: amos
Subject: trigonometric

Using 18 degrees as x, prove that

sin 54 - sin 18 = 1/2

and

sin 54 * sin 18 = 1/4

I have tried proving it but I can't go beyond

(cos2x - sinx)/2

Because 18 is exactly one-third of 54, I know that

sin 54 - sin 18 = 2cos 2x sin x.
cos 2x = sin 3x
2sin 3x sin x = -(cos 4x - cos 2x)
cos 4x = sin x
sin 3x sin x = (cos 2x - sin x)/2

Help me out. Thank you.

```

```
Date: 04/01/2010 at 13:02:55
From: Doctor Robert
Subject: Re: trigonometric

Hello,

What is the original problem? I don't see why the cos is brought into play
when you are asked to solve

sin54 - sinx = 1/4

sin54 - 1/4 = sinx

sin x = .5590

x = 33.99 degrees

- Doctor Robert, The Math Forum
http://mathforum.org/dr.math/

```

```
Date: 04/13/2010 at 12:31:05
From: amos
Subject: trigonometric

The problem is not to solve, it is to PROVE that

sin 54 - sin 18 = 1/2

Thank you very much.

```

```
Date: 04/15/2010 at 10:00:47
From: Doctor Jacques
Subject: Re: trigonometric

Hi Amos,

You want to prove that:

sin 54 - sin 18 = 1/2
sin 54 * sin 18 = 1/4

First, let us show that the two propositions are equivalent. Use the
formula:

sin p - sin q = 2*sin ((p - q)/2) cos ((p + q)/2)

with p = 54.

- Doctor Jacques, The Math Forum
http://mathforum.org/dr.math/

```

```
Date: 04/16/2010 at 04:25:12
From: amos
Subject: trigonometric

Please can you go further solving this question?

sin 72 = 2sin 36 sin 54,

I keep mixing it up.

Thank you.

```

```
Date: 04/16/2010 at 06:23:21
From: Doctor Jacques
Subject: Re: trigonometric

Hi Amos,

Once you find that, you can substitute sin 72 into equation [2]

sin 36 = 2 sin 18 sin 72                 [2]

This substitution gives

sin 36 = 2 sin 18 (2 sin 36 sin 54)
= 4 sin 18 sin 36 sin 54          [4]

And, if you divide both sides of [4] by (4 sin 36), you obtain:

1/4 = sin 18 sin 54

This is the second equation, which we wanted to prove. As explained
before, the first equation is a consequence of it.

Does this make sense?

- Doctor Jacques, The Math Forum
http://mathforum.org/dr.math/

```

```
Date: 04/27/2010 at 08:49:30
From: amos
Subject: Thank you (trigonometric)

Thank you so much. I am so grateful. You really helped me. God bless you.
```
Associated Topics:
High School Trigonometry

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