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
1o 0.0175 0.9998 2o 0.0349 0.9994
3o 0.0523 0.9986 4o 0.0698 0.9976
5o 0.0872 0.9962 6o 0.1045 0.9945
7o 0.1219 0.9925 8o 0.1392 0.9903
9o 0.1564 0.9877 10o 0.1736 0.9848
20o 0.3420 0.9397 30o 0.5000 0.8660
40o 0.6428 0.7660

Example 1: Find sin(45o).

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(45o).

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 40o. That's because we can take advantage of the fact that sin(90) = 1 and cos(90) = 0.

Example 3: Find sin(75o).

sin(75) = sin(90 - 15)
= sin(90)cos(15) + cos(90)sin(15)
= cos(15)

Example 4: Find cos(75o).

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 15o and 75o. 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.

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