Addition formula for sine and for cosine (C)

It more difficult to see that there are identities for the sine and cosine of the sum of two angles:

$$\cos(A+B) = \cos(A)\cos(B)-\sin(A)\sin(B)$$

and

$$\sin(A+B) = \sin(A)\cos(B) + \cos(A)\sin(B).$$

Exercises:

1.

If $\cos(A) = 3/5$, $\sin(A) = 4/5$, $\cos(B) = 5/13$ and $\sin(\theta) = 12/13$ find the sine and cosine of A+B.

2.

If $\cos(A) = -3/5$, $\sin(A) = 4/5$, $\cos(B) = 5/13$ and $\sin(\theta) = -12/13$ find the sine and cosine of A+B.

3.

Show that if $h\neq 0$ then

$$\frac{\cos(A+h)-\cos(A)}{h} = -\sin(A)\frac{\sin(h)}{h} -\cos(A)\frac{1-\cos(h)}{h}$$

and

$$\frac{\sin(A+h)-\sin(A)}{h} = \cos(A)\frac{\sin(h)}{h} +\sin(A)\frac{1-\cos(h)}{h}.$$

4.

Using the odd and even properties of sine and cosine, derive the subtraction identities

$$\cos(A-B) = \cos(A)\cos(B) + \sin(A)\sin(B)$$

and

$$\sin(A-B) = \sin(A)\cos(B) - \cos(A)\sin(B).$$

This shows that the subtraction identities should not be committed to memory.

5.

If $\cos(A) = 3/5$, $\sin(A) = 4/5$, $\cos(B) = 5/13$ and $\sin(\theta) = 12/13$ find the sine and cosine of A-B.

6.

If $\cos(A) = -3/5$, $\sin(A) = 4/5$, $\cos(B) = 5/13$ and $\sin(\theta) = -12/13$ find the sine and cosine of A-B.