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

By Murray Bourne, 23 Jun 2011

**Challenge: **What is the exact value for sine of 6 degrees? How about sine of 1 degree?

**Context: **I received a delightful email from reader James Parent recently. He wrote:

I have the exact answers for the sin of all integer angles. Has anyone done this before? I'm retired, and a Professor Emeritus from a community college. I'm 74 years-old.

This certainly sounded interesting to me, so I asked James to write a guest post, and here it is. (Many of James' mails had the tag-line "Sent from my iPad".)

Over to James.

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

Here is one way of going about it.

**Background**

Let’s find some **exact values** using some well-known triangles. Then we’ll use these exact values to answer the above challenges.

**sin 45****°: **You may recall that an isosceles right triangle with sides of 1 and with hypotenuse of square root of 2 will give you the sine of 45 degrees as half the square root of 2.

**sin 30****° and sin 60****°: **An equilateral triangle has all angles measuring 60 degrees and all three sides are equal. For convenience, we choose each side to be length 2. When you bisect an angle, you get 30 degrees and the side opposite is 1/2 of 2, which gives you 1. Using that right triangle, you get exact answers for sine of 30°, and sin 60° which are 1/2 and the square root of 3 over 2 respectively.

## Using these results – sine 15°

How do you find the value of the sine of 15°?

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

So the sine of 1/2 of 30° will be:

which gives us

or

**Note: **We could also find the sine of 15 degrees using sine (45° − 30°).

**sin 75****°**: Now using the formula for the sine of the sum of 2 angles,

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

we can find the sine of (45° + 30°) to give sine of 75 degrees.

We now find the sine of 36°, by first finding the cos of 36°.

**cos 36****°: **The cosine of 36 degrees can be calculated by using a pentagon. See cos36° at CutTheKnot where it is shown that

Putting these values on a right triangle and solving for the unknown side, we can conclude:

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

Calculating this, the sine of 18 degrees becomes

**sin 3****°: **The above leads you to one of the paths to sine of 3 degrees and to sine of 6 degrees.

For example, sine (18° - 15°) will give us the sine of 3 degrees. which is

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

This gives us the following value of sin 3°:

or other forms depending how you factor the above.

**sin 6****°****: **Using the above, one can compute the sine of 6 degrees finally as sine of twice 3 degrees to arrive at

**sin 18****° and sin 72****°:** Taking the equivalent sine and cosine values of 15° and 18° on the right hand side of

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

gives us:

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

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

**Sines of other angles**

Many angles can be computed exactly by many methods. Another practical formula is the sine of 3 times an angle:

sin 3*A* = 3 sin *A* − 4 sin^{3}*A*

**sin 9****°: **For example, the sine of 9 degrees is the sine of (3×3°).

So, with A = 3, we arrive at

And so on.

**sin 1****°: **Now, to find the sine of one degree, one needs to know sine of one third of three degrees!

One needs to solve the above for sin (A) in terms of 3A, and this involves solving the cubic. As you know, the cubic was solved many, many years ago.

There are three solutions and one needs to know which one to use and when! Experience has taught me to use the following for a quadrant I angle (the "*I*" in this expression stands for the imaginary number √(−1). See Complex Numbers for more information.)

[Click image to see full size]

Use the following when you have a quadrant II angle:

Use the following for quadrant III angles:

[Click image to see full size]

So, the expression for sine(1°) becomes

[Click image to see full size]

Messy, isn't it! But, it does give you the exact value for the sine of one degree.

## Is it correct?

Evaluate the sine of 1 degree using a TI Scientific Calculator and you will get 0.0174524064. Evaluate the above messy expression and you will also get 0.0174524064. Even allowing for calculator rounding errors, we can be confident our answer is correct.

## List of all sines of integer degrees from 1° to 90°

This PDF contains all the exact values of the sine values for whole-numbered angles (in degrees):

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

**Concluding Comments from James **

For a retired community college mathematics professor since 1997, this has been a lot of enjoyment for me.

James Parent, Professor Emeritus

Schenectady County Community College, Schenectady, New York

Currently teaching as an adjunct at Great Bay Community College, Portsmouth, New Hampshire

## Alternative List: sines and cosines of integer degrees from 3° to 90° (multiples of 3)

Reader Herwig Ronsmans wrote a comment to say he had found the sines and cosines of multiples of 3° (i.e. 3°, 6°, 9°, ...). He mentioned his list has simpler expressions for many of the angles compared to the above PDF.

He kindly shared his list: Sines and cosines of multiples of 3° [PDF, 165 kB].

It's in Dutch, but as Herwig said, mathematics is universal and so it's easy to work out what he's doing if you're not a Dutch speaker.

## Simplifying Radicals

As pointed out by several commenters and readers, this solution is lengthy for the sake of completeness, but can be reduced in steps using the Simplest Radical Form.

See the 102 Comments below.