Question

find the exact value of the expression. do not use a calculator.

\\( \sin \left(70^{\circ}\
ight) \cos \left(20^{\circ}\
ight)+\sin \left(20^{\circ}\
ight) \cos \left(70^{\circ}\
ight) \\)

\\( \sin \left(70^{\circ}\
ight) \cos \left(20^{\circ}\
ight)+\sin \left(20^{\circ}\
ight) \cos \left(70^{\circ}\
ight)= \\)
(simplify your answer, including any radicals. use integers or fractions for any numbers ir
as needed.)

Explanation:

Step1: Use the sine addition formula

The formula for \(\sin(A + B)=\sin A\cos B+\cos A\sin B\).
In the given expression \(\sin(70^{\circ})\cos(20^{\circ})+\sin(20^{\circ})\cos(70^{\circ})\), let \(A = 70^{\circ}\) and \(B=20^{\circ}\). Then the expression becomes \(\sin(70^{\circ}+ 20^{\circ})\).

Step2: Simplify the angle

\(70^{\circ}+20^{\circ}=90^{\circ}\), so the expression is \(\sin(90^{\circ})\).

Step3: Evaluate the sine function

We know that \(\sin(90^{\circ}) = 1\).

Answer:

\(1\)