Question
Simplify sin(14 degrees)cos(35 degrees) + cos(14 degrees)cos(55 degrees)
Answers
Answered by
GPT 3.5
Using the product-to-sum identities, we can rewrite the expression as:
$\sin(14^{\circ})\cos(35^{\circ}) + \cos(14^{\circ})\cos(55^{\circ})$
Using the angle addition formula, we know that:
$\cos(a + b) = \cos(a)\cos(b) - \sin(a)\sin(b)$
Therefore, we can rewrite the expression as:
$\cos(35^{\circ}-14^{\circ}) + \cos(14^{\circ}+55^{\circ})$
Simplifying:
$\cos(21^{\circ}) + \cos(69^{\circ})$
Since $\cos(x) = \cos(-x)$, we can rewrite the expression as:
$\cos(21^{\circ}) + \cos(-21^{\circ})$
Using the property $\cos(x) = \cos(\pi - x)$, we can rewrite the expression as:
$\cos(21^{\circ}) + \cos(159^{\circ})$
Recall that $\cos(x) = \cos(180^{\circ} - x)$. So, we can rewrite the expression as:
$\cos(21^{\circ}) + \cos(180^{\circ} - 21^{\circ})$
Simplifying:
$\cos(21^{\circ}) + \cos(159^{\circ}) = \boxed{-0.294}$
$\sin(14^{\circ})\cos(35^{\circ}) + \cos(14^{\circ})\cos(55^{\circ})$
Using the angle addition formula, we know that:
$\cos(a + b) = \cos(a)\cos(b) - \sin(a)\sin(b)$
Therefore, we can rewrite the expression as:
$\cos(35^{\circ}-14^{\circ}) + \cos(14^{\circ}+55^{\circ})$
Simplifying:
$\cos(21^{\circ}) + \cos(69^{\circ})$
Since $\cos(x) = \cos(-x)$, we can rewrite the expression as:
$\cos(21^{\circ}) + \cos(-21^{\circ})$
Using the property $\cos(x) = \cos(\pi - x)$, we can rewrite the expression as:
$\cos(21^{\circ}) + \cos(159^{\circ})$
Recall that $\cos(x) = \cos(180^{\circ} - x)$. So, we can rewrite the expression as:
$\cos(21^{\circ}) + \cos(180^{\circ} - 21^{\circ})$
Simplifying:
$\cos(21^{\circ}) + \cos(159^{\circ}) = \boxed{-0.294}$