Find the value of 3 sin 15° - 4 sin 315°.

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Accepted Answer

Step 1: Calculate sin 15°.
Using the formula sin(15°) = sin(45° - 30°) = sin 45° cos 30° - cos 45° sin 30°.
We know sin 45° = √2/2, cos 30° = √3/2, cos 45° = √2/2, sin 30° = 1/2.
So, sin 15° = (√2/2)(√3/2) - (√2/2)(1/2) = (√6/4) - (√2/4) = (√6 - √2)/4.

Step 2: Calculate sin 315°.
315° = 360° - 45°, so sin 315° = -sin 45° = -√2/2.

Step 3: Substitute values in the expression 3 sin 15° - 4 sin 315°.
= 3 * ((√6 - √2)/4) - 4 * (-√2/2)
= (3(√6 - √2))/4 + 4 * (√2/2)
= (3√6 - 3√2)/4 + 2√2
= (3√6 - 3√2)/4 + (8√2)/4
= (3√6 - 3√2 + 8√2)/4
= (3√6 + 5√2)/4

Step 4: Approximate values.
√6 ≈ 2.449, √2 ≈ 1.414
So, (3*2.449 + 5*1.414)/4 = (7.347 + 7.07)/4 = 14.417/4 ≈ 3.604

This does not match any options directly, so let's re-check calculations.

Re-examining sin 15°:
sin 15° = (√6 - √2)/4 ≈ (2.449 - 1.414)/4 = 1.035/4 = 0.2587
3 sin 15° = 3 * 0.2587 = 0.776

sin 315° = -√2/2 ≈ -0.707
-4 sin 315° = -4 * (-0.707) = 2.828

Sum = 0.776 + 2.828 = 3.604

Since none of the options match 3.604, let's check if the question might have a typo or if the options correspond to a different interpretation.

Alternatively, check if sin 315° is taken as positive (which is incorrect).

If sin 315° = sin(-45°) = -sin 45° = -√2/2, so the calculation is correct.

Hence, the value is approximately 3.6, which is closest to option 1 (2) but not exact.

Double-check the problem statement or options.

Alternatively, check if the problem meant cos 315° instead of sin 315°.

cos 315° = cos(360° - 45°) = cos 45° = √2/2 ≈ 0.707

Then expression: 3 sin 15° - 4 cos 315° = 0.776 - 4*0.707 = 0.776 - 2.828 = -2.052 (no match).

Conclusion: The exact value of the expression is (3√6 + 5√2)/4 ≈ 3.604, which does not match the given options exactly.

Therefore, the closest option is 2, but the exact value is as calculated.

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