& How do you find exact values for the sine of all angles?
How do you find exact values for the sine of all angles?
By , 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
which gives us
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
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
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 (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 (18& & 15&) = sin 18& cos 15& & sin 15& cos 18&
sin 18& sin 75& & sin 15& sin 72&
We can calculate the values of the
sines of 18& and
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 3A = 3 sin A & 4 sin3A
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
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.. Evaluate the above messy expression and you will also get 0.. 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):
[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,
Currently teaching as an adjunct at
Community College, Portsmouth, New Hampshire
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23 Jun 2011 []
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观察下列等式：2-12＝8＝8×152-32＝16＝8×272-52＝24＝8×392-72＝32＝8×4（1）若a2-b2=8×11，则a=______，b=______．（2）根据上述规律，第n个等式是______．
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（1）∵a2-b2=8×11，∴a=2×11+1=23，b=2×11-1=21；（2）32-12=（2×1+1）2-（2×1-1）2=8=8×1，52-32=（2×2+1）2-（2×2-1）2=16=8×2，72-52=（2×3+1）2-（2×3-1）2=24=8×3，92-72=（2×4+1）2-（2×4-1）2=32=8×4，…第n个等式是（2n+1）2-（2n-1）2=8n．故答案为：（1）23，21；（2）（2n+1）2-（2n-1）2=8n．
为您推荐：
（1）观察规律不难发现，等号左边的两个底数分别是与8相乘的因数的2倍加1与2倍减1，然后求出a、b即可；（2）根据左边是奇数列的平方的差，右边是8与相应的等式的序号的积，写出即可．
本题考点：
规律型：数字的变化类．
考点点评：
本题是对数字变化规律的考查，观察得出等号左边的两个底数分别是与8相乘的因数的2倍加1与2倍减1是解题的关键，也是本题的难点，此类题目对同学们的能力要求较高．
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我有更好的答案
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冒泡应该可以！
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