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How do you find exact values for the sine of all angles?

By Murray Bourne, 23 Jun 2011

Problem: What’s the actual worth for sine of 6 levels? How about sine of 1 diploma?

Context: I obtained a pleasant e mail from reader James Father or mother lately. He wrote:

I’ve the precise solutions for the sin of all integer angles. Has anybody executed this earlier than? I am retired, and a Professor Emeritus from a group school. I am 74 years-old.

This actually sounded fascinating to me, so I requested James to write down a visitor put up, and right here it’s. (Lots of James’ mails had the tag-line “Sent from my iPad”.)

Over to James.

How do you discover actual values for the sine of integer angles?

Right here is a technique of going about it.

Background

Let’s discover some actual values utilizing some well-known triangles. Then we’ll use these actual values to reply the above challenges.

sin 45°: You might recall that an isosceles proper triangle with sides of 1 and with hypotenuse of sq. root of two provides you with the sine of 45 levels as half the sq. root of two.

sin 45

sin 30° and sin 60°: An equilateral triangle has all angles measuring  60 levels and all three sides are equal.  For comfort, we select either side to be size 2.  Once you bisect an angle, you get 30 levels and the aspect reverse is 1/2 of two, which supplies you 1.  Utilizing that proper triangle, you get actual solutions for sine of 30°, and sin 60° that are 1/2 and the sq. root of three over 2 respectively.

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Utilizing these outcomes – sine 15°

How do you discover the worth of the sine of 15°?  

Sine of half an angle within the first quadrant is given by the expression:

So the sine of 1/2 of 30° can be:

which supplies us

or

Word: We may additionally discover the sine of 15 levels utilizing sine (45° − 30°).

sin 75°: Now utilizing the components for the sine of the sum of two angles,

sin(A + B) = sin A cos B + cos A sin B,

we are able to discover the sine of (45° + 30°) to provide sine of 75 levels.

We now discover the sine of 36°, by first discovering the cos of 36°.

cos 36°: The cosine of 36 levels will be calculated through the use of a pentagon. See cos36° at CutTheKnot the place it’s proven that

cos36^o=frac{1+sqrt{5}}{4}

Placing these values on a proper triangle and fixing for the unknown aspect, we are able to conclude:

sin36^o=frac{sqrt{10-2sqrt{5}}}{4}

sin 18°: Now, the sine of 18 levels comes from the sine of half of 36 levels.

Calculating this, the sine of 18 levels turns into

sin 3°: The above leads you to one of many paths to sine of three levels and to sine of 6 levels.

For instance, sine (18° – 15°) will give us the sine of three levels. which is

sin 3° = sin (18° − 15°) = sin 18° cos 15° − sin 15° cos 18°

This provides us the next worth of sin 3°:

or different types relying the way you issue the above. 

sin 6°: Utilizing the above, one can compute the sine of 6 levels lastly as  sine of twice 3 levels to reach at

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sin 18° and sin 72°: Taking the equal sine and cosine values of 15° and 18° on the precise hand aspect of

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sin 3° = sin (18° − 15°) = sin 18° cos 15° − sin 15° cos 18°

offers us:

sin 3° = sin 18° sin 75° − sin 15° sin 72°

We will calculate the values of the sines of 18° and 72° from the above expression.

Sines of different angles
Many angles will be computed precisely by many strategies.  One other sensible components is the sine of three instances an angle:

sin 3 = 3 sin − 4 sin3

sin 9°: For instance, the sine of 9 levels is the sine of (3×3°). 

So, with A = 3, we arrive at

And so forth.

sin 1°: Now, to search out the sine of 1 diploma, one must know sine of 1 third of three levels!

One wants to resolve the above for sin (A) when it comes to 3A, and this entails fixing the cubic.  As you understand, the cubic was solved many, a few years in the past.

There are three options and one must know which one to make use of and when! Expertise has taught me to make use of the next for a quadrant I angle (the “I” on this expression stands for the imaginary quantity √(−1). See Advanced Numbers for extra info.)


[Click image to see full size]

Use the next when you could have a quadrant II angle:

Use the next for quadrant III angles:


[Click image to see full size]

So, the expression for sine(1°) turns into

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[Click image to see full size]

Messy, is not it! However, it does provide the actual worth for the sine of 1 diploma.

Is it right?

Consider the sine of 1 diploma utilizing a TI Scientific Calculator and you’re going to get 0.0174524064. Consider the above messy expression and additionally, you will get 0.0174524064. Even permitting for calculator rounding errors, we will be assured our reply is right.

Listing of all sines of integer levels from 1° to 90°

This PDF incorporates all the precise values of the sine values for whole-numbered angles (in levels):

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Precise values sin 1° to sin 90° [PDF, 293 kB]

Concluding Feedback from James

For a retired group school arithmetic professor since 1997, this has been lots of enjoyment for me.

James Father or mother, Professor Emeritus
Schenectady County Neighborhood Faculty, Schenectady, New York
At the moment educating as an adjunct at Nice Bay Neighborhood Faculty, Portsmouth, New Hampshire

Different Listing: sines and cosines of integer levels from 3° to 90° (multiples of three)

Reader Herwig Ronsmans wrote a remark to say he had discovered the sines and cosines of multiples of three° (i.e. 3°, 6°, 9°, …). He talked about his record has easier expressions for lots of the angles in comparison with the above PDF.

He kindly shared his record: Sines and cosines of multiples of three° [PDF, 165 kB].

It is in Dutch, however as Herwig stated, arithmetic is common and so it is easy to work out what he is doing when you’re not a Dutch speaker.

 

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