Given that sin(𝐴 + 𝐵) = sin 𝐴 cos𝐵 +
cos 𝐴 sin𝐵, find the value of sin 105 °.
A. √42 (√3 + 1)
B. √42 (√3 − 1)
C. √42 (1 − √3)
D. √6 + √2
We know that sin(A+B) = sinAcosB + cosAsinB. We can use this formula for A = 60° and B = 45° as follows:
sin(60° + 45°) = sin60°cos45° + cos60°sin45°
Simplifying using known values:
sin(105°) = (√3/2)(√2/2) + (1/2)(√2/2)
sin(105°) = (√6 + √2)/4
Therefore, the answer is D, √6 + √2.
To find the value of sin 105°, we can use the formula sin(𝐴 + 𝐵) = sin 𝐴 cos𝐵 + cos 𝐴 sin𝐵.
Let's express 105° as a sum of two angles: 105° = 60° + 45°.
Using the formula, we have:
sin(105°) = sin(60° + 45°)
= sin(60°)cos(45°) + cos(60°)sin(45°)
Now, we know that sin 60° = √3/2, cos 60° = 1/2, sin 45° = √2/2, and cos 45° = √2/2.
Substituting these values into the formula, we get:
sin(105°) = (√3/2)(√2/2) + (1/2)(√2/2)
= (√3√2)/4 + √2/4
= (√6 + √2)/4
= (√6 + √2)/2√2
To rationalize the denominator, we multiply the numerator and denominator by √2:
= [(√6 + √2)/2√2] * (√2/√2)
= (√12 + 2)/4√2
= (√12 + 2)/4√2 * (√2/√2)
= (√24 + 4√2)/8
= (√24 + 4√2)/8
We notice that √24 = √(4 * 6) = 2√6.
Therefore, we have:
sin(105°) = (2√6 + 4√2)/8
= √6/4 + √2/2
= √6/4 + (2/2)√2
= √6/4 + √2/1
= (√6 + 4√2)/4
= (1/4)(√6 + 4√2)
Comparing this result with the given options, we find that the answer is:
A. √42 (√3 + 1)