Trig Sum Forumulas
Algebra: Trigonometry Help
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First of all we start with the formulas.
sin(a + b) =
sin(a)cos(b) + cos(a)sin(b)
sin(a - b) =
sin(a)cos(b) - cos(a)sin(b)
cos(a + b) =
cos(a)cos(b) - sin(a)sin(b)
cos(a - b) =
cos(a)cos(b) + sin(a)sin(b)
Next, we give you a table of sine and cosine values.
Angle | sin | cos | Angle | sin | cos |
---|---|---|---|---|---|
1^{o} | 0.0175 | 0.9998 | 2^{o} | 0.0349 | 0.9994 |
3^{o} | 0.0523 | 0.9986 | 4^{o} | 0.0698 | 0.9976 |
5^{o} | 0.0872 | 0.9962 | 6^{o} | 0.1045 | 0.9945 |
7^{o} | 0.1219 | 0.9925 | 8^{o} | 0.1392 | 0.9903 |
9^{o} | 0.1564 | 0.9877 | 10^{o} | 0.1736 | 0.9848 |
20^{o} | 0.3420 | 0.9397 | 30^{o} | 0.5000 | 0.8660 |
40^{o} | 0.6428 | 0.7660 |
Example 1: Find sin(45^{o}).
sin(45) = sin(40 + 5)
= sin(40)cos(5) + cos(40)sin(5)
= (0.6428)(0.9962) + (0.766)(0.9872)
= 0.7071
Example 2: Find cos(45^{o}).
cos(45) = cos(40 + 5)
= cos(40)cos(5) - sin(40)sin(5)
= (0.766)(0.9962) - (0.6428)(0.9872)
= 0.7071
The observant ones among you would have noticed that the angles only go up to 40^{o}. That's because we can take advantage of the fact that sin(90) = 1 and cos(90) = 0.
Example 3: Find sin(75^{o}).
sin(75) = sin(90 - 15)
= sin(90)cos(15) + cos(90)sin(15)
= cos(15)
Example 4: Find cos(75^{o}).
cos(75) = cos(90 - 15)
= cos(90)cos(15) + sin(90)sin(15)
= sin(15)
This should be apparent if you draw a right-angle triangle with the angles of 15^{o} and 75^{o}. If you need a refresher on the basics of trig, check out the basic trigonometry facts page.
There are also trig identities involving the sum angle formulas.
Algebra: Trigonometry Help
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