cos(a+b) and sin(a+b) theorm
Answers
Step-by-step explanation:
As the examples showed, sometimes we need angles other than 0, 30, 45, 60, and 90 degrees. In this chapter you need to learn two things:
1. Sin(A + B) is not equal to sin A + sin B. It doesn't work like removing the parentheses in algebra.
2. The formula for what sin(A + B) does equal.
First to show that removing parentheses doesn't "work." Here: make A 30 degrees and B 45 degrees.
Sin 30 is 0.5. Sin 45 is 0.7071. Adding the two is 1.2071.
You know that no sine (or cosine) can be more than 1. Why? the ratio has the hypotenuse as its denominator. The most that the numerator can be is equal to the denominator. A sine or cosine can never be greater than 1, so a value of 1.2071 must be wrong.
Finding sin(A + B)
The easiest way to find sin(A + B), uses the geometrical construction shown here. The big angle, (A + B), consists of two smaller ones, A and B, The construction (1) shows that the opposite side is made of two parts. The lower part, divided by the line between the angles (2), is sin A. The line between the two angles divided by the hypotenuse (3) is cos B. Multiply the two together. The middle line is in both the numerator and denominator, so each cancels and leaves the lower part of the opposite over the hypotenuse (4).
sin(A + B)
Notice the little right triangle (5). The shaded angle is A, because the line on its top side is parallel to the base line. Similar right triangles with an angle A show that the top angle, marked A, also equals the original A. The top part of the opposite (6), over the longest of that shaded triangle, is cos A. The opposite over the main hypotenuse (7) is sin B. Since the side marked "opposite" (7) is in both the numerator and denominator when cos A and sin B are multiplied together, cos A sin B is the top part of the original opposite — for (A + B) — divided by the main hypotenuse (8).
Now, put it all together (9). Sin(A + B) is the two parts of the opposite - all divided by the hypotenuse (9). Putting that into its trig form:
sin(A + B) = sin A cos B + cos A sin B
Finding cos(A + B)
A very similar construction finds the formula for the cosine of an angle made with two angles added together.
cos(A + B)
Using the same construction (1), notice that the adjacent side is the full base line (for cos A), with part of it subtracted at the right. Each part must use the same denominator, the hypotenuse of the (A + B) triangle.
The full base line, divided by the dividing line between angles A and E, is cos A (2). This dividing line, divided by the hypotenuse of (A + B) triangle, is cos B (3). So, the full base line divided by the hypotenuse is the product cos A cos B (4).
Now, for the little part that has to be subtracted. The shaded part (5) represents sin A, which multiplied by the shaded part (6) is sin E, which produces the other piece you need (7). The subtraction produces cos(A + B) (8) so that the formula we need is:
cos(A + B) = cos A cos B - sin A sin B
sina sinb = 1. 2(cos(a − b) − cos(a + b))