Trigonometric functions of sum and difference of two angles

The governing concept is the sine addition identity, sin(A + B) = sin A cos B + cos A sin B. Put A = π and B = θ. Then sin(π + θ) = sin π cos θ + cos π sin θ. Since sin π = 0 and cos π = −1, the expression becomes 0·cos θ + (−1)·sin θ = −sin θ. Therefore option A is correct. The unit-circle interpretation gives the same result: adding π makes a half-turn, moving a point to the diametrically opposite point and reversing its y-coordinate, which represents sine. Option B overlooks this sign reversal, while options C and D incorrectly replace the y-coordinate function with cosine.

Use the cosine addition identity, cos(A + B) = cos A cos B − sin A sin B. With A = π and B = θ, cos(π + θ) = cos π cos θ − sin π sin θ. The standard values are cos π = −1 and sin π = 0, so the expression becomes (−1)cos θ − 0·sin θ = −cos θ. Hence option B is correct. Geometrically, adding π produces a half-turn on the unit circle. A point with coordinates (cos θ, sin θ) moves to (−cos θ, −sin θ), so its x-coordinate, represented by cosine, changes sign. Option A misses that reversal, whereas options C and D confuse cosine with the y-coordinate function, sine.

Direct answer: √3/2, so option D is correct. Recall the sine-difference identity: sin(A − B) = sin A cos B − cos A sin B. The given expression has exactly this pattern, with A = 75° and B = 15°. Therefore it becomes sin(75° − 15°) = sin 60°. From the standard special-angle values, sin 60° = √3/2. Option D is therefore the only correct choice. Option A, 0, would be sin 0°, but the angle difference is 60°, not 0°. Option B, 1/2, is sin 30°, so it corresponds to the wrong angle. Option C, 1, is sin 90°, which also does not match. A useful warning is not to change the order of subtraction: sin(A − B) has the exact sign pattern shown in the question. Memory cue: ‘sine first angle times cosine second, minus cosine first times sine second’ means sine of the difference.