Question

write the expression as the sine, cosine, or tangent of an angle. then find the exact value of the expression.
\\( \sin 20 ^ { \circ } \cos 10 ^ { \circ } + \cos 20 ^ { \circ } \sin 10 ^ { \circ } \\)
write the expression as the sine, cosine, or tangent of an angle.
\\( \sin 20 ^ { \circ } \cos 10 ^ { \circ } + \cos 20 ^ { \circ } \sin 10 ^ { \circ } = \square ^ { \circ } \\)
(type an exact answer, using \\( \pi \\) as needed. use integers or fractions for any numbers in the expression.)
find the exact value of the expression.
\\( \sin 20 ^ { \circ } \cos 10 ^ { \circ } + \cos 20 ^ { \circ } \sin 10 ^ { \circ } = \square \\)
(simplify your answer. type an exact answer, using radicals as needed. use integers or fractions for any numbers in the
expression. rationalize all denominators.)

Explanation:

Step1: Apply the sine addition formula

The formula for \(\sin(A + B)=\sin A\cos B+\cos A\sin B\).
Here \(A = 20^{\circ}\) and \(B=10^{\circ}\), so \(\sin20^{\circ}\cos10^{\circ}+\cos20^{\circ}\sin10^{\circ}=\sin(20^{\circ}+ 10^{\circ})\).

Step2: Calculate the angle sum

\(20^{\circ}+10^{\circ}=30^{\circ}\), so the expression is \(\sin30^{\circ}\).

Step3: Find the exact value

We know that \(\sin30^{\circ}=\frac{1}{2}\).

Answer:

\(\sin20^{\circ}\cos10^{\circ}+\cos20^{\circ}\sin10^{\circ}=\sin30^{\circ}\) and \(\sin20^{\circ}\cos10^{\circ}+\cos20^{\circ}\sin10^{\circ}=\frac{1}{2}\)