6. Given that sin(𝐴 + 𝐵) = sin 𝐴 cos𝐵 +

Question

cos 𝐴 sin𝐵, find the value of sin 105 °.
A. √42 (√3 + 1)
B. √42 (√3 − 1)
C. √42 (1 − √3)
D. √6 + √2

Answer 1

We can use the identity sin(𝐴 + 𝐵) = sin 𝐴 cos𝐵 + cos 𝐴 sin𝐵 to find sin 105°. First, we note that 105° can be expressed as the sum of two angles, 60° and 45°:

105° = 60° + 45°

Using the identity, we can write:

sin 105° = sin(60° + 45°) = sin 60° cos 45° + cos 60° sin 45°

We know that sin 60° = √3/2 and cos 45° = sin 45° = 1/√2 and cos 60° = 1/2, so we can substitute these values:

sin 105° = (√3/2)(1/√2) + (1/2)(1/√2) = (√3 + 1)/2√2

We can simplify this expression by multiplying both numerator and denominator by √2:

sin 105° = (√3 + 1)/2√2 x (√2/√2) = (√6 + √2)/4

Therefore, the answer is (D) √6 + √2.