Sin 12 degree sin 18 degree sin 54 degree
The Correct Answer and Explanation is:
To evaluate the expression:sin(12∘)⋅sin(18∘)⋅sin(54∘)\sin(12^\circ) \cdot \sin(18^\circ) \cdot \sin(54^\circ)sin(12∘)⋅sin(18∘)⋅sin(54∘)
we can make use of known trigonometric identities and exact values for special angles.
Step-by-step calculation:
Start by using known sine values for 18° and 54°:
- sin(18∘)=5−14\sin(18^\circ) = \dfrac{\sqrt{5} – 1}{4}sin(18∘)=45−1
- sin(54∘)=5+14\sin(54^\circ) = \dfrac{\sqrt{5} + 1}{4}sin(54∘)=45+1
Multiplying these two:sin(18∘)⋅sin(54∘)=(5−14)⋅(5+14)\sin(18^\circ) \cdot \sin(54^\circ) = \left( \dfrac{\sqrt{5} – 1}{4} \right) \cdot \left( \dfrac{\sqrt{5} + 1}{4} \right)sin(18∘)⋅sin(54∘)=(45−1)⋅(45+1)
Use the identity (a−b)(a+b)=a2−b2(a – b)(a + b) = a^2 – b^2(a−b)(a+b)=a2−b2:=5−116=416=14= \dfrac{5 – 1}{16} = \dfrac{4}{16} = \dfrac{1}{4}=165−1=164=41
Now, multiply that by sin(12∘)\sin(12^\circ)sin(12∘):sin(12∘)⋅14\sin(12^\circ) \cdot \dfrac{1}{4}sin(12∘)⋅41
We now need the value of sin(12∘)\sin(12^\circ)sin(12∘). This does not have a simple radical form, but using a calculator or table:sin(12∘)≈0.2079\sin(12^\circ) \approx 0.2079sin(12∘)≈0.2079
Therefore:sin(12∘)⋅sin(18∘)⋅sin(54∘)≈0.2079⋅14=0.2079⋅0.25=0.051975\sin(12^\circ) \cdot \sin(18^\circ) \cdot \sin(54^\circ) \approx 0.2079 \cdot \dfrac{1}{4} = 0.2079 \cdot 0.25 = 0.051975sin(12∘)⋅sin(18∘)⋅sin(54∘)≈0.2079⋅41=0.2079⋅0.25=0.051975
Rounded to four decimal places:0.0520\boxed{0.0520}0.0520
Explanation
The given expression involves multiplying three sine values: sin(12∘)\sin(12^\circ)sin(12∘), sin(18∘)\sin(18^\circ)sin(18∘), and sin(54∘)\sin(54^\circ)sin(54∘). These are not standard angles typically found on the unit circle, but some of them, such as 18° and 54°, are linked to the geometry of regular pentagons and the golden ratio. Their exact values can be expressed using square roots.
First, we use known exact values:
- sin(18∘)\sin(18^\circ)sin(18∘) equals (5−1)/4(\sqrt{5} – 1)/4(5−1)/4
- sin(54∘)\sin(54^\circ)sin(54∘) equals (5+1)/4(\sqrt{5} + 1)/4(5+1)/4
When multiplied, these two expressions form a difference of squares:(5−14)(5+14)=5−116=14\left(\dfrac{\sqrt{5} – 1}{4}\right)\left(\dfrac{\sqrt{5} + 1}{4}\right) = \dfrac{5 – 1}{16} = \dfrac{1}{4}(45−1)(45+1)=165−1=41
This reduces our problem to finding the product of sin(12∘)\sin(12^\circ)sin(12∘) and 1/41/41/4. While sin(12∘)\sin(12^\circ)sin(12∘) does not have a simple radical form, it can be approximated numerically using a calculator. The value is about 0.2079. Multiplying by 1/4 gives approximately 0.0520.
This method demonstrates the power of using identities and known exact values to simplify seemingly complex trigonometric products. The final result is a simple decimal, even though the original expression involved non-standard angles.
