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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

Letâ€™s discover some actual values utilizing some well-known triangles. Then weâ€™ll use these actual values to reply the above challenges.

sin 45Â°: 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 30Â° and sin 60Â°: An equilateral triangle has all angles measuringÂ  60 levels and all three sides are equal.Â  For comfort, we select either side to be size 2.Â  Once you bisect an angle, you get 30 levels and the aspect reverse is 1/2 of two, which supplies you 1.Â  Utilizing that proper triangle, you get actual solutions for sine of 30Â°, and sin 60Â° that are 1/2 and the sq. root of three over 2 respectively.

## Utilizing these outcomes â€“ sine 15Â°

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

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

So the sine of 1/2 of 30Â° can be:

which supplies us

or

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

sin 75Â°: 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 (45Â° + 30Â°) to provide sine of 75 levels.

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

cos 36Â°: The cosine of 36 levels will be calculated through the use of a pentagon. See cos36Â° at CutTheKnot the place it’s proven that

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

sin 18Â°: 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 3Â°: The above leads you to one of many paths to sine of three levels and to sine of 6 levels.

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

sin 3Â° = sin (18Â° âˆ’ 15Â°) = sin 18Â° cos 15Â° âˆ’ sin 15Â° cos 18Â°

This provides us the next worth of sin 3Â°:

or different types relying the way you issue the above.Â

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

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

sin 3Â° = sin (18Â° âˆ’ 15Â°) = sin 18Â° cos 15Â° âˆ’ sin 15Â° cos 18Â°

offers us:

sin 3Â° = sin 18Â° sin 75Â° âˆ’ sin 15Â° sin 72Â°

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

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

sin 3 = 3 sin âˆ’ 4 sin3

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

So, with A = 3, we arrive at

And so forth.

sin 1Â°: 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.Â  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 âˆš(âˆ’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(1Â°) turns into

[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 1Â° to 90Â°

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

Precise values sin 1Â° to sin 90Â° [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
At the moment educating as an adjunct at Nice Bay Neighborhood Faculty, Portsmouth, New Hampshire

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

Reader Herwig Ronsmans wrote a remark to say he had discovered the sines and cosines of multiples of threeÂ° (i.e. 3Â°, 6Â°, 9Â°, …). 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 threeÂ° [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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