Sin84 sin48 sin24 sin12= 1/4*(2sin84 sin24) (2sin48 sin12)= 1/4*(cos60- cos108)(cos36- cos60)= 1/4*(1/2 + sin18) (cos36 - 1/2)= 1/4*(1/2 + (root5 - 1)/4) ((root5 + 1)/4 - 1/2)= 1/4* ((root5 + 1)/4) ((root5 - 1)/4)= 1/64 * (5-1)= 1/16 Formula used: cosA - cosB = 2sin((a+b)/2) sin((a-b)/2)Values of sin18 = (root5 - 1)/4 and cos36 = (root5 + 1)/4

Sin 12°sin24°sin48°sin84°=

To solve the equation Sin 12°sin24°sin48°sin84°, we need to break it down step by step.

Step 1: Convert degrees to radians
The trigonometric functions in most calculators and mathematical formulas use radians instead of degrees. Therefore, we need to convert the given angles from degrees to radians.

Angle in radians = (Angle in degrees * π) / 180

Let's convert the given angles to radians:

12° = (12 * π) / 180 = 0.2094 radians
24° = (24 * π) / 180 = 0.4189 radians
48° = (48 * π) / 180 = 0.8378 radians
84° = (84 * π) / 180 = 1.4661 radians

Step 2: Evaluate the sin of each angle
Now that we have the angles in radians, we can find the sine of each angle using a calculator or mathematical tables. Here are the values:

sin 0.2094 ≈ 0.2079
sin 0.4189 ≈ 0.4132
sin 0.8378 ≈ 0.7445
sin 1.4661 ≈ 0.9953

Step 3: Multiply the sine values
Finally, we multiply the sine values of all four angles together to get the result:

sin 12°sin24°sin48°sin84° ≈ 0.2079 * 0.4132 * 0.7445 * 0.9953 ≈ 0.0645

Therefore, sin 12°sin24°sin48°sin84° ≈ 0.0645.

Conclusion:
The value of sin 12°sin24°sin48°sin84° is approximately 0.0645.

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