# Question 366e9

##### 2 Answers
Feb 2, 2018

$\sin \left(10 \theta\right) + \sin \left(4 \theta\right)$

#### Explanation:

start with the sum and difference formulas for sine:
$\sin \left(a + b\right) = \sin \left(a\right) \cos \left(b\right) + \cos \left(a\right) \sin \left(b\right)$
$\sin \left(a - b\right) = \sin \left(a\right) \cos \left(b\right) - \cos \left(a\right) \sin \left(b\right)$

add them together to get:

$\sin \left(a + b\right) + \sin \left(a - b\right) = 2 \sin \left(a\right) \cos \left(b\right)$

For this problem, let $a = 7 \theta$ and $b = 3 \theta$, so

$2 \sin \left(7 \theta\right) \cos \left(3 \theta\right) = \sin \left(7 \theta + 3 \theta\right) + \sin \left(7 \theta - 3 \theta\right)$
$= \sin \left(10 \theta\right) + \sin \left(4 \theta\right)$

Feb 2, 2018

$\sin 10 \theta + \sin 4 \theta$

#### Explanation:

$\text{using the "color(blue)"product to sum formula}$

•color(white)(x)2sinAcosB=sin(A+B)+sin(A-B)#

$\text{here "A=7theta" and } B = 3 \theta$

$\Rightarrow 2 \sin 7 \theta \cos 3 \theta$

$= \sin \left(7 \theta + 3 \theta\right) + \sin \left(7 \theta - 3 \theta\right)$

$= \sin 10 \theta + \sin 4 \theta$