6. 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 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.
To find the value of sin 105° using the given trigonometric identity, we need to express 105° as the sum of two angles.
Since 105° is not a special angle, we can express it as the sum of 90° and 15°:
105° = 90° + 15°
Now, we can use the given trigonometric identity sin(A + B) = sin A cos B + cos A sin B.
Let A = 90° and B = 15°:
sin 105° = sin(90° + 15°) = sin 90° cos 15° + cos 90° sin 15°
We know that sin 90° = 1 and cos 90° = 0, so substituting these values, we have:
sin 105° = 1 * cos 15° + 0 * sin 15°
sin 105° = cos 15°
Now, we need to find the value of cos 15°.
Using the half-angle formula cos 2θ = 1 - 2sin^2 θ, we can express cos 15° as:
cos 15° = cos(30° / 2)
cos 15° = √((1 + cos 30°) / 2)
Since cos 30° = √3 / 2, we can substitute this value:
cos 15° = √((1 + √3 / 2) / 2)
cos 15° = √((2 + √3) / 4)
cos 15° = √(2 + √3) / 2
Now, we have found the value of cos 15°. However, we need to find sin 105°, so we can use the Pythagorean identity sin^2 θ + cos^2 θ = 1.
Since sin^2 θ + cos^2 θ = 1, we can solve for sin θ:
sin^2 θ = 1 - cos^2 θ
sin θ = √(1 - cos^2 θ)
Plugging in the value of cos 15°, we have:
sin 105° = √(1 - (√(2 + √3) / 2)^2)
sin 105° = √(1 - (2 + √3) / 4)
sin 105° = √(4 - 2 - √3) / 4
sin 105° = √(2 - √3) / 2
Therefore, the value of sin 105° is √(2 - √3) / 2.
None of the options A, B, C, or D match this value, so none of the options is correct.