Question

8 cos 10° × cos 50° × cos 70°​

Answer 1

Решение задания приложено

Answer 2

To find the value of the expression 8 cos 10° × cos 50° × cos 70°, you can use the trigonometric identity:

cos(A) × cos(B) = (1/2) * [cos(A - B) + cos(A + B)]

Applying this identity, we can rewrite the expression as follows:

8 cos 10° × cos 50° × cos 70° = 8 * (1/2) * [cos(50° - 10°) + cos(50° + 10°)] × cos 70° = 4 * [cos 40° + cos 60°] × cos 70°

Now, we can further simplify using the same identity:

cos(A) × cos(B) = (1/2) * [cos(A - B) + cos(A + B)]

cos 40° + cos 60° = (1/2) * [cos(40° - 60°) + cos(40° + 60°)] = (1/2) * [cos(-20°) + cos(100°)]

Since cos(-θ) = cos(θ), we can rewrite the expression as:

(1/2) * [cos(-20°) + cos(100°)] = (1/2) * [cos(20°) + cos(100°)]

Finally, substituting this back into the original expression:

4 * [cos 40° + cos 60°] × cos 70° = 4 * [(1/2) * [cos(20°) + cos(100°)]] × cos 70°

At this point, we have simplified the expression as much as possible without using an approximation.